Properties

Degree 2
Conductor $ 5^{2} \cdot 7 \cdot 23 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.11·2-s + 1.84·3-s + 2.49·4-s − 3.90·6-s − 7-s − 1.04·8-s + 0.388·9-s + 5.87·11-s + 4.58·12-s + 6.24·13-s + 2.11·14-s − 2.77·16-s + 5.42·17-s − 0.823·18-s − 2.23·19-s − 1.84·21-s − 12.4·22-s + 23-s − 1.92·24-s − 13.2·26-s − 4.80·27-s − 2.49·28-s + 0.642·29-s + 7.84·31-s + 7.96·32-s + 10.8·33-s − 11.4·34-s + ⋯
L(s)  = 1  − 1.49·2-s + 1.06·3-s + 1.24·4-s − 1.59·6-s − 0.377·7-s − 0.368·8-s + 0.129·9-s + 1.77·11-s + 1.32·12-s + 1.73·13-s + 0.566·14-s − 0.693·16-s + 1.31·17-s − 0.193·18-s − 0.513·19-s − 0.401·21-s − 2.65·22-s + 0.208·23-s − 0.391·24-s − 2.59·26-s − 0.925·27-s − 0.470·28-s + 0.119·29-s + 1.40·31-s + 1.40·32-s + 1.88·33-s − 1.97·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4025} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.626030094$
$L(\frac12)$  $\approx$  $1.626030094$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good2 \( 1 + 2.11T + 2T^{2} \)
3 \( 1 - 1.84T + 3T^{2} \)
11 \( 1 - 5.87T + 11T^{2} \)
13 \( 1 - 6.24T + 13T^{2} \)
17 \( 1 - 5.42T + 17T^{2} \)
19 \( 1 + 2.23T + 19T^{2} \)
29 \( 1 - 0.642T + 29T^{2} \)
31 \( 1 - 7.84T + 31T^{2} \)
37 \( 1 + 0.557T + 37T^{2} \)
41 \( 1 - 2.56T + 41T^{2} \)
43 \( 1 - 8.81T + 43T^{2} \)
47 \( 1 + 4.26T + 47T^{2} \)
53 \( 1 + 3.01T + 53T^{2} \)
59 \( 1 - 4.17T + 59T^{2} \)
61 \( 1 + 0.148T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + 7.93T + 71T^{2} \)
73 \( 1 + 4.28T + 73T^{2} \)
79 \( 1 + 0.861T + 79T^{2} \)
83 \( 1 - 4.81T + 83T^{2} \)
89 \( 1 - 6.32T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.633157579130651555193273340765, −8.037235925054931713347312035055, −7.30839693812742425778902220768, −6.40750528881524051740652765395, −5.96493131231868019457246844792, −4.31339139682242642499576689476, −3.61292867385496940735518057142, −2.81679252097921039293356143089, −1.59302858438911053648773445340, −0.976402381734085407172900394593, 0.976402381734085407172900394593, 1.59302858438911053648773445340, 2.81679252097921039293356143089, 3.61292867385496940735518057142, 4.31339139682242642499576689476, 5.96493131231868019457246844792, 6.40750528881524051740652765395, 7.30839693812742425778902220768, 8.037235925054931713347312035055, 8.633157579130651555193273340765

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