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  − 2.69·2-s + 0.269·3-s + 5.23·4-s − 0.725·6-s − 7-s − 8.70·8-s − 2.92·9-s − 3.78·11-s + 1.41·12-s − 1.24·13-s + 2.69·14-s + 12.9·16-s − 5.98·17-s + 7.87·18-s − 3.38·19-s − 0.269·21-s + 10.1·22-s + 23-s − 2.34·24-s + 3.33·26-s − 1.59·27-s − 5.23·28-s − 7.02·29-s + 6.26·31-s − 17.4·32-s − 1.02·33-s + 16.1·34-s + ⋯
L(s)  = 1  − 1.90·2-s + 0.155·3-s + 2.61·4-s − 0.296·6-s − 0.377·7-s − 3.07·8-s − 0.975·9-s − 1.14·11-s + 0.407·12-s − 0.344·13-s + 0.718·14-s + 3.23·16-s − 1.45·17-s + 1.85·18-s − 0.775·19-s − 0.0588·21-s + 2.17·22-s + 0.208·23-s − 0.479·24-s + 0.654·26-s − 0.307·27-s − 0.989·28-s − 1.30·29-s + 1.12·31-s − 3.08·32-s − 0.177·33-s + 2.76·34-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  =  $(2,\ 4025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1709285813$
$L(\frac12)$  $\approx$  $0.1709285813$
$L(\frac{3}{2})$   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 \( 1 + 2.69T + 2T^{2} \)
3 \( 1 - 0.269T + 3T^{2} \)
11 \( 1 + 3.78T + 11T^{2} \)
13 \( 1 + 1.24T + 13T^{2} \)
17 \( 1 + 5.98T + 17T^{2} \)
19 \( 1 + 3.38T + 19T^{2} \)
29 \( 1 + 7.02T + 29T^{2} \)
31 \( 1 - 6.26T + 31T^{2} \)
37 \( 1 + 4.84T + 37T^{2} \)
41 \( 1 + 1.78T + 41T^{2} \)
43 \( 1 + 3.02T + 43T^{2} \)
47 \( 1 + 3.90T + 47T^{2} \)
53 \( 1 - 2.47T + 53T^{2} \)
59 \( 1 + 2.89T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 - 2.70T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 - 3.31T + 79T^{2} \)
83 \( 1 + 7.02T + 83T^{2} \)
89 \( 1 + 1.59T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−8.559239407409102059138369609578, −7.948513044486693948890164492568, −7.24629647226574559624412982872, −6.50212477576003919237924776957, −5.88389878524314177802605112099, −4.82327374701349192620961318618, −3.32154930869442125169186800964, −2.53370932152700804856023121755, −1.92843561491559684228218792398, −0.28680288490728902721378591750, 0.28680288490728902721378591750, 1.92843561491559684228218792398, 2.53370932152700804856023121755, 3.32154930869442125169186800964, 4.82327374701349192620961318618, 5.88389878524314177802605112099, 6.50212477576003919237924776957, 7.24629647226574559624412982872, 7.948513044486693948890164492568, 8.559239407409102059138369609578

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