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Properties

Label 4-2376e2-1.1-c1e2-0-15
Degree $4$
Conductor $5645376$
Sign $-1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
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  + 3·5-s − 3·11-s − 6·23-s − 3·25-s + 3·31-s − 2·37-s − 3·47-s + 49-s + 24·53-s − 9·55-s + 9·59-s − 67-s − 12·71-s − 24·89-s − 97-s + 25·103-s + 9·113-s − 18·115-s − 2·121-s − 30·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 9·155-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.904·11-s − 1.25·23-s − 3/5·25-s + 0.538·31-s − 0.328·37-s − 0.437·47-s + 1/7·49-s + 3.29·53-s − 1.21·55-s + 1.17·59-s − 0.122·67-s − 1.42·71-s − 2.54·89-s − 0.101·97-s + 2.46·103-s + 0.846·113-s − 1.67·115-s − 0.181·121-s − 2.68·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.722·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
11$C_2$ \( 1 + 3 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.5.ad_m
7$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.7.a_ab
13$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.13.a_f
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.17.a_g
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.23.g_t
29$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.29.a_d
31$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ad_ck
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.c_bn
41$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.41.a_t
43$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \) 2.43.a_ch
47$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.d_cc
53$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) 2.53.ay_jp
59$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.aj_fi
61$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.61.a_abf
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.b_ek
71$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.m_gn
73$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.73.a_aw
79$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \) 2.79.a_ch
83$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.83.a_ax
89$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.y_mg
97$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.b_hk
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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

−7.08452405971432940799301204244, −6.65011538932975859687230764994, −6.01114120567852398815280354125, −5.83068772378892851851679788104, −5.63235841300855216029006093840, −5.13494060614288839810291853271, −4.63830435893640473479868574085, −4.09258197719311291941799448914, −3.76911520310687424036271775379, −3.08561999762785588427184855936, −2.50912618311590876940043568843, −2.17491699535189382450309558436, −1.78077113179979885899411711697, −0.969030002666050280194242099200, 0, 0.969030002666050280194242099200, 1.78077113179979885899411711697, 2.17491699535189382450309558436, 2.50912618311590876940043568843, 3.08561999762785588427184855936, 3.76911520310687424036271775379, 4.09258197719311291941799448914, 4.63830435893640473479868574085, 5.13494060614288839810291853271, 5.63235841300855216029006093840, 5.83068772378892851851679788104, 6.01114120567852398815280354125, 6.65011538932975859687230764994, 7.08452405971432940799301204244

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