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Properties

Label 4-100352-1.1-c1e2-0-24
Degree $4$
Conductor $100352$
Sign $-1$
Analytic cond. $6.39853$
Root an. cond. $1.59045$
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  − 3·7-s + 4·9-s − 6·17-s − 14·23-s − 2·25-s + 2·31-s − 8·41-s − 12·47-s + 2·49-s − 12·63-s + 13·71-s − 6·73-s − 4·79-s + 7·81-s + 12·89-s + 15·97-s + 19·103-s − 22·113-s + 18·119-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + ⋯
L(s)  = 1  − 1.13·7-s + 4/3·9-s − 1.45·17-s − 2.91·23-s − 2/5·25-s + 0.359·31-s − 1.24·41-s − 1.75·47-s + 2/7·49-s − 1.51·63-s + 1.54·71-s − 0.702·73-s − 0.450·79-s + 7/9·81-s + 1.27·89-s + 1.52·97-s + 1.87·103-s − 2.06·113-s + 1.65·119-s + 1.09·121-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 − 1.94·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100352\)    =    \(2^{11} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(6.39853\)
Root analytic conductor: \(1.59045\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 100352,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
7$C_2$ \( 1 + 3 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.11.a_am
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.13.a_i
17$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.g_bb
19$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.19.a_abe
23$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.23.o_dn
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.ac_cc
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.37.a_c
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.i_de
43$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.43.a_ac
47$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.m_ek
53$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.53.a_abc
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.59.a_ack
61$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.61.a_bo
67$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.67.a_adq
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.71.an_fy
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.g_fy
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.e_s
83$C_2^2$ \( 1 - 36 T^{2} + p^{2} T^{4} \) 2.83.a_abk
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.am_fu
97$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.ap_im
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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

−9.461629738850377944537427359853, −8.932478334846632090726123698795, −8.228641958772684175794965602224, −7.920905830332773209065548494163, −7.24662133618947414268649191652, −6.68816649600718255343604577993, −6.34735324219322177806737128498, −5.97856645934258042172649280775, −5.00660257013163213679742776368, −4.47689845934523097951470131925, −3.86776710950254674265913343859, −3.44804166921272108753166731405, −2.32640521625712662014207077275, −1.74438294267145940336829441245, 0, 1.74438294267145940336829441245, 2.32640521625712662014207077275, 3.44804166921272108753166731405, 3.86776710950254674265913343859, 4.47689845934523097951470131925, 5.00660257013163213679742776368, 5.97856645934258042172649280775, 6.34735324219322177806737128498, 6.68816649600718255343604577993, 7.24662133618947414268649191652, 7.920905830332773209065548494163, 8.228641958772684175794965602224, 8.932478334846632090726123698795, 9.461629738850377944537427359853

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