Properties

Label 2-4025-1.1-c1-0-155
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.744·2-s + 1.85·3-s − 1.44·4-s − 1.38·6-s − 7-s + 2.56·8-s + 0.456·9-s − 4.15·11-s − 2.68·12-s + 4.70·13-s + 0.744·14-s + 0.980·16-s + 7.26·17-s − 0.339·18-s − 7.26·19-s − 1.85·21-s + 3.09·22-s − 23-s + 4.77·24-s − 3.50·26-s − 4.72·27-s + 1.44·28-s + 6.03·29-s − 6.07·31-s − 5.86·32-s − 7.71·33-s − 5.40·34-s + ⋯
L(s)  = 1  − 0.526·2-s + 1.07·3-s − 0.722·4-s − 0.565·6-s − 0.377·7-s + 0.907·8-s + 0.152·9-s − 1.25·11-s − 0.775·12-s + 1.30·13-s + 0.199·14-s + 0.245·16-s + 1.76·17-s − 0.0801·18-s − 1.66·19-s − 0.405·21-s + 0.658·22-s − 0.208·23-s + 0.973·24-s − 0.687·26-s − 0.910·27-s + 0.273·28-s + 1.12·29-s − 1.09·31-s − 1.03·32-s − 1.34·33-s − 0.927·34-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: \(1\)
Selberg data: \((2,\ 4025,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.744T + 2T^{2} \)
3 \( 1 - 1.85T + 3T^{2} \)
11 \( 1 + 4.15T + 11T^{2} \)
13 \( 1 - 4.70T + 13T^{2} \)
17 \( 1 - 7.26T + 17T^{2} \)
19 \( 1 + 7.26T + 19T^{2} \)
29 \( 1 - 6.03T + 29T^{2} \)
31 \( 1 + 6.07T + 31T^{2} \)
37 \( 1 + 2.81T + 37T^{2} \)
41 \( 1 + 9.74T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + 9.52T + 47T^{2} \)
53 \( 1 - 2.72T + 53T^{2} \)
59 \( 1 - 0.216T + 59T^{2} \)
61 \( 1 + 5.26T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 1.44T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 + 3.55T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 - 2.63T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−8.033930320076029631632889539809, −7.977317579895782067403367855689, −6.78294664932564115649141448976, −5.77704239129482246939938054632, −5.11628387620869404051842669633, −3.95074176918271592424230656121, −3.45026393411626924748189102926, −2.51070847492956363538780725615, −1.39112088980006605823938700058, 0, 1.39112088980006605823938700058, 2.51070847492956363538780725615, 3.45026393411626924748189102926, 3.95074176918271592424230656121, 5.11628387620869404051842669633, 5.77704239129482246939938054632, 6.78294664932564115649141448976, 7.977317579895782067403367855689, 8.033930320076029631632889539809

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