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Properties

Label 4-1050e2-1.1-c1e2-0-5
Degree $4$
Conductor $1102500$
Sign $1$
Analytic cond. $70.2963$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s + 16-s − 16·19-s − 12·29-s − 8·31-s + 36-s − 12·41-s − 49-s + 24·59-s − 20·61-s − 64-s + 24·71-s + 16·76-s − 16·79-s + 81-s + 12·89-s + 12·101-s − 28·109-s + 12·116-s − 22·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1/4·16-s − 3.67·19-s − 2.22·29-s − 1.43·31-s + 1/6·36-s − 1.87·41-s − 1/7·49-s + 3.12·59-s − 2.56·61-s − 1/8·64-s + 2.84·71-s + 1.83·76-s − 1.80·79-s + 1/9·81-s + 1.27·89-s + 1.19·101-s − 2.68·109-s + 1.11·116-s − 2·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1102500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(70.2963\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1102500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4087677406\)
\(L(\frac12)\) \(\approx\) \(0.4087677406\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.19.q_dy
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−10.37204067895360918481983228832, −9.473874051663781037138254672848, −9.359851012353017973039367266267, −8.780923119626631225513626118080, −8.597280079766162540289583543151, −8.078473581600251918966968228864, −7.84219075785560575448812641665, −7.00031212581663480313945020426, −6.86542985455919831086181389273, −6.26295393848384490139988056849, −5.92239313671592521064508708772, −5.24932878668138570319052657656, −5.09154103786102797841496572143, −4.24589462716121997227799806971, −3.93806844338426601468012027860, −3.63973708419947515253285945118, −2.77021536865459198252462738341, −1.90277964016715099523092025671, −1.90095491420799327919340864741, −0.27291155692953918357030306629, 0.27291155692953918357030306629, 1.90095491420799327919340864741, 1.90277964016715099523092025671, 2.77021536865459198252462738341, 3.63973708419947515253285945118, 3.93806844338426601468012027860, 4.24589462716121997227799806971, 5.09154103786102797841496572143, 5.24932878668138570319052657656, 5.92239313671592521064508708772, 6.26295393848384490139988056849, 6.86542985455919831086181389273, 7.00031212581663480313945020426, 7.84219075785560575448812641665, 8.078473581600251918966968228864, 8.597280079766162540289583543151, 8.780923119626631225513626118080, 9.359851012353017973039367266267, 9.473874051663781037138254672848, 10.37204067895360918481983228832

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