Properties

Label 2-4025-1.1-c1-0-93
Degree $2$
Conductor $4025$
Sign $1$
Analytic cond. $32.1397$
Root an. cond. $5.66919$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.78·2-s − 0.0742·3-s + 1.19·4-s − 0.132·6-s + 7-s − 1.43·8-s − 2.99·9-s + 4.34·11-s − 0.0887·12-s + 5.31·13-s + 1.78·14-s − 4.96·16-s + 2.25·17-s − 5.35·18-s + 2.09·19-s − 0.0742·21-s + 7.76·22-s − 23-s + 0.106·24-s + 9.49·26-s + 0.444·27-s + 1.19·28-s − 8.32·29-s + 3.81·31-s − 5.99·32-s − 0.322·33-s + 4.03·34-s + ⋯
L(s)  = 1  + 1.26·2-s − 0.0428·3-s + 0.597·4-s − 0.0541·6-s + 0.377·7-s − 0.508·8-s − 0.998·9-s + 1.30·11-s − 0.0256·12-s + 1.47·13-s + 0.477·14-s − 1.24·16-s + 0.547·17-s − 1.26·18-s + 0.480·19-s − 0.0161·21-s + 1.65·22-s − 0.208·23-s + 0.0217·24-s + 1.86·26-s + 0.0856·27-s + 0.225·28-s − 1.54·29-s + 0.684·31-s − 1.05·32-s − 0.0560·33-s + 0.692·34-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

Degree: \(2\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(32.1397\)
Root analytic conductor: \(5.66919\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.846751836\)
\(L(\frac12)\) \(\approx\) \(3.846751836\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 - 1.78T + 2T^{2} \)
3 \( 1 + 0.0742T + 3T^{2} \)
11 \( 1 - 4.34T + 11T^{2} \)
13 \( 1 - 5.31T + 13T^{2} \)
17 \( 1 - 2.25T + 17T^{2} \)
19 \( 1 - 2.09T + 19T^{2} \)
29 \( 1 + 8.32T + 29T^{2} \)
31 \( 1 - 3.81T + 31T^{2} \)
37 \( 1 - 3.34T + 37T^{2} \)
41 \( 1 - 0.389T + 41T^{2} \)
43 \( 1 - 2.40T + 43T^{2} \)
47 \( 1 + 0.385T + 47T^{2} \)
53 \( 1 - 9.38T + 53T^{2} \)
59 \( 1 + 7.25T + 59T^{2} \)
61 \( 1 - 8.11T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 - 3.78T + 73T^{2} \)
79 \( 1 - 5.30T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 18.7T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.564450150384013057543449766981, −7.62883431655135095967421696216, −6.60931153628496633671293562173, −5.96470244239739818253359361316, −5.57757222362214781591777506847, −4.62749405665831549897715318354, −3.72336615249092472991978150838, −3.42884314800694963558913232865, −2.22540941458293592365041621615, −0.972131520870796013393978200253, 0.972131520870796013393978200253, 2.22540941458293592365041621615, 3.42884314800694963558913232865, 3.72336615249092472991978150838, 4.62749405665831549897715318354, 5.57757222362214781591777506847, 5.96470244239739818253359361316, 6.60931153628496633671293562173, 7.62883431655135095967421696216, 8.564450150384013057543449766981

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