Properties

Label 2-4025-1.1-c1-0-102
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  − 0.515·2-s + 3.36·3-s − 1.73·4-s − 1.73·6-s − 7-s + 1.92·8-s + 8.30·9-s + 1.20·11-s − 5.83·12-s + 2.81·13-s + 0.515·14-s + 2.47·16-s − 1.95·17-s − 4.28·18-s − 0.107·19-s − 3.36·21-s − 0.623·22-s + 23-s + 6.47·24-s − 1.45·26-s + 17.8·27-s + 1.73·28-s + 0.846·29-s − 8.74·31-s − 5.12·32-s + 4.06·33-s + 1.00·34-s + ⋯
L(s)  = 1  − 0.364·2-s + 1.94·3-s − 0.867·4-s − 0.707·6-s − 0.377·7-s + 0.680·8-s + 2.76·9-s + 0.364·11-s − 1.68·12-s + 0.780·13-s + 0.137·14-s + 0.618·16-s − 0.473·17-s − 1.00·18-s − 0.0247·19-s − 0.733·21-s − 0.132·22-s + 0.208·23-s + 1.32·24-s − 0.284·26-s + 3.43·27-s + 0.327·28-s + 0.157·29-s − 1.57·31-s − 0.906·32-s + 0.707·33-s + 0.172·34-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

\(L(1)\) \(\approx\) \(2.915940300\)
\(L(\frac12)\) \(\approx\) \(2.915940300\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.515T + 2T^{2} \)
3 \( 1 - 3.36T + 3T^{2} \)
11 \( 1 - 1.20T + 11T^{2} \)
13 \( 1 - 2.81T + 13T^{2} \)
17 \( 1 + 1.95T + 17T^{2} \)
19 \( 1 + 0.107T + 19T^{2} \)
29 \( 1 - 0.846T + 29T^{2} \)
31 \( 1 + 8.74T + 31T^{2} \)
37 \( 1 - 2.07T + 37T^{2} \)
41 \( 1 + 0.748T + 41T^{2} \)
43 \( 1 - 9.08T + 43T^{2} \)
47 \( 1 - 8.58T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + 4.49T + 59T^{2} \)
61 \( 1 - 1.14T + 61T^{2} \)
67 \( 1 + 0.821T + 67T^{2} \)
71 \( 1 - 2.02T + 71T^{2} \)
73 \( 1 + 7.95T + 73T^{2} \)
79 \( 1 - 7.88T + 79T^{2} \)
83 \( 1 + 0.403T + 83T^{2} \)
89 \( 1 - 6.13T + 89T^{2} \)
97 \( 1 - 18.0T + 97T^{2} \)
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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

−8.734706228748942844716978266573, −7.83464965946116970043708688619, −7.39617121147745836784400373469, −6.49898409423235683973067532130, −5.33332238024382813341342625499, −4.10487991707008634799709634304, −3.95400525474373600227272260158, −2.99884006898263828269258512739, −2.02605779941841189664988870961, −1.01040981636889861289167822214, 1.01040981636889861289167822214, 2.02605779941841189664988870961, 2.99884006898263828269258512739, 3.95400525474373600227272260158, 4.10487991707008634799709634304, 5.33332238024382813341342625499, 6.49898409423235683973067532130, 7.39617121147745836784400373469, 7.83464965946116970043708688619, 8.734706228748942844716978266573

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