Properties

Label 4-1274098-1.1-c1e2-0-0
Degree $4$
Conductor $1274098$
Sign $-1$
Analytic cond. $81.2375$
Root an. cond. $3.00219$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4-s − 7-s − 2·8-s − 2·9-s − 6·11-s − 2·14-s − 3·16-s − 4·18-s − 12·22-s + 10·23-s − 4·25-s − 28-s + 3·29-s + 2·32-s − 2·36-s + 7·43-s − 6·44-s + 20·46-s − 6·49-s − 8·50-s + 7·53-s + 2·56-s + 6·58-s + 2·63-s + 9·64-s + 26·67-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 0.377·7-s − 0.707·8-s − 2/3·9-s − 1.80·11-s − 0.534·14-s − 3/4·16-s − 0.942·18-s − 2.55·22-s + 2.08·23-s − 4/5·25-s − 0.188·28-s + 0.557·29-s + 0.353·32-s − 1/3·36-s + 1.06·43-s − 0.904·44-s + 2.94·46-s − 6/7·49-s − 1.13·50-s + 0.961·53-s + 0.267·56-s + 0.787·58-s + 0.251·63-s + 9/8·64-s + 3.17·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274098 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274098 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1274098\)    =    \(2 \cdot 7^{2} \cdot 13001\)
Sign: $-1$
Analytic conductor: \(81.2375\)
Root analytic conductor: \(3.00219\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1274098,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_2$ \( ( 1 - T )( 1 - T + p T^{2} ) \)
7$C_2$ \( 1 + T + p T^{2} \)
13001$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 198 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.g_w
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.17.a_p
19$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.19.a_ae
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.23.ak_cs
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.29.ad_cc
31$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.31.a_q
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.41.a_abh
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.43.ah_do
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.53.ah_eo
59$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.59.a_abs
61$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.61.a_abu
67$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.67.aba_li
71$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.c_ade
73$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.73.a_ai
79$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.79.g_fv
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.89.a_de
97$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.97.a_bl
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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

−7.917920583050504175488621616611, −7.17818009149322705132409761680, −6.78072522608441334092300185794, −6.36907134054194114237231705154, −5.69174002799585150523946450094, −5.48061732560703696442827764483, −5.15034086081698216889443988136, −4.74571228437771025575073976602, −4.16900834522139684975520486534, −3.62148410687285398919974749906, −3.14086290800127605143633193338, −2.59911843805746534254207815627, −2.42993400567805687489983894453, −1.04107594015981333403442849676, 0, 1.04107594015981333403442849676, 2.42993400567805687489983894453, 2.59911843805746534254207815627, 3.14086290800127605143633193338, 3.62148410687285398919974749906, 4.16900834522139684975520486534, 4.74571228437771025575073976602, 5.15034086081698216889443988136, 5.48061732560703696442827764483, 5.69174002799585150523946450094, 6.36907134054194114237231705154, 6.78072522608441334092300185794, 7.17818009149322705132409761680, 7.917920583050504175488621616611

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