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Properties

Label 4-1134e2-1.1-c1e2-0-60
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·5-s + 5·7-s + 8-s − 3·10-s + 3·11-s + 10·13-s − 5·14-s − 16-s − 3·17-s − 5·19-s − 3·22-s + 3·23-s + 5·25-s − 10·26-s − 6·29-s + 4·31-s + 3·34-s + 15·35-s + 7·37-s + 5·38-s + 3·40-s − 18·41-s + 22·43-s − 3·46-s + 18·49-s − 5·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.34·5-s + 1.88·7-s + 0.353·8-s − 0.948·10-s + 0.904·11-s + 2.77·13-s − 1.33·14-s − 1/4·16-s − 0.727·17-s − 1.14·19-s − 0.639·22-s + 0.625·23-s + 25-s − 1.96·26-s − 1.11·29-s + 0.718·31-s + 0.514·34-s + 2.53·35-s + 1.15·37-s + 0.811·38-s + 0.474·40-s − 2.81·41-s + 3.35·43-s − 0.442·46-s + 18/7·49-s − 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.189497559\)
\(L(\frac12)\) \(\approx\) \(3.189497559\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.13.ak_bz
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_g
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_m
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.41.s_gh
43$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.43.aw_hz
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_abs
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_acf
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.73.l_bw
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.83.ag_gt
89$C_2^2$ \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.89.p_fg
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.97.c_hn
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.985413541299328583687788790165, −9.531839825282041212728865143932, −8.897557826754265740611833831606, −8.858249025202094601361118251707, −8.526470214405757806620769444914, −8.330505763909598493291323750498, −7.42880160330321178648643420657, −7.41505524817667468855690408781, −6.41135792438382055680359099547, −6.38066635759790739748394604443, −5.80642262808500219174224670333, −5.57626934615906133306985603133, −4.74194032930466284981216847931, −4.42531156920292385387827249016, −4.00718857618165977103205232663, −3.38205928983064962954836194584, −2.41227160379637542816328091597, −1.91619097732881201945307798283, −1.35141061699251876538048749812, −1.09292580925615169553171066545, 1.09292580925615169553171066545, 1.35141061699251876538048749812, 1.91619097732881201945307798283, 2.41227160379637542816328091597, 3.38205928983064962954836194584, 4.00718857618165977103205232663, 4.42531156920292385387827249016, 4.74194032930466284981216847931, 5.57626934615906133306985603133, 5.80642262808500219174224670333, 6.38066635759790739748394604443, 6.41135792438382055680359099547, 7.41505524817667468855690408781, 7.42880160330321178648643420657, 8.330505763909598493291323750498, 8.526470214405757806620769444914, 8.858249025202094601361118251707, 8.897557826754265740611833831606, 9.531839825282041212728865143932, 9.985413541299328583687788790165

Graph of the $Z$-function along the critical line