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  + 1.50·2-s + 3.04·3-s + 0.256·4-s + 4.56·6-s − 7-s − 2.61·8-s + 6.24·9-s − 0.542·11-s + 0.779·12-s + 1.21·13-s − 1.50·14-s − 4.44·16-s + 3.66·17-s + 9.38·18-s + 5.00·19-s − 3.04·21-s − 0.814·22-s + 23-s − 7.96·24-s + 1.82·26-s + 9.86·27-s − 0.256·28-s + 3.72·29-s + 9.04·31-s − 1.44·32-s − 1.64·33-s + 5.50·34-s + ⋯
L(s)  = 1  + 1.06·2-s + 1.75·3-s + 0.128·4-s + 1.86·6-s − 0.377·7-s − 0.925·8-s + 2.08·9-s − 0.163·11-s + 0.225·12-s + 0.337·13-s − 0.401·14-s − 1.11·16-s + 0.888·17-s + 2.21·18-s + 1.14·19-s − 0.663·21-s − 0.173·22-s + 0.208·23-s − 1.62·24-s + 0.358·26-s + 1.89·27-s − 0.0484·28-s + 0.692·29-s + 1.62·31-s − 0.254·32-s − 0.286·33-s + 0.943·34-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  =  $(2,\ 4025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.859323121$
$L(\frac12)$  $\approx$  $5.859323121$
$L(\frac{3}{2})$   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 \( 1 - 1.50T + 2T^{2} \)
3 \( 1 - 3.04T + 3T^{2} \)
11 \( 1 + 0.542T + 11T^{2} \)
13 \( 1 - 1.21T + 13T^{2} \)
17 \( 1 - 3.66T + 17T^{2} \)
19 \( 1 - 5.00T + 19T^{2} \)
29 \( 1 - 3.72T + 29T^{2} \)
31 \( 1 - 9.04T + 31T^{2} \)
37 \( 1 - 9.08T + 37T^{2} \)
41 \( 1 + 4.86T + 41T^{2} \)
43 \( 1 + 3.64T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + 7.48T + 53T^{2} \)
59 \( 1 - 5.18T + 59T^{2} \)
61 \( 1 - 3.26T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 5.29T + 71T^{2} \)
73 \( 1 + 8.19T + 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 7.64T + 83T^{2} \)
89 \( 1 + 4.74T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−8.260703831083219701229827901622, −7.982282059982500729278874242974, −6.93359635079501138574458857353, −6.24328367032104703271232964821, −5.20886954584608056373707479197, −4.49966034321092593205799779576, −3.62424003567001716283429204942, −3.11881927544921843405647070031, −2.56790214007102384837815112914, −1.18483011812085461729716526493, 1.18483011812085461729716526493, 2.56790214007102384837815112914, 3.11881927544921843405647070031, 3.62424003567001716283429204942, 4.49966034321092593205799779576, 5.20886954584608056373707479197, 6.24328367032104703271232964821, 6.93359635079501138574458857353, 7.982282059982500729278874242974, 8.260703831083219701229827901622

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