Properties

Degree $2$
Conductor $6025$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.930·2-s + 1.32·3-s − 1.13·4-s − 1.23·6-s − 4.33·7-s + 2.91·8-s − 1.24·9-s + 0.882·11-s − 1.50·12-s + 2.68·13-s + 4.03·14-s − 0.442·16-s + 0.304·17-s + 1.15·18-s − 4.13·19-s − 5.74·21-s − 0.820·22-s − 2.19·23-s + 3.86·24-s − 2.49·26-s − 5.62·27-s + 4.91·28-s + 8.97·29-s + 5.91·31-s − 5.41·32-s + 1.17·33-s − 0.283·34-s + ⋯
L(s)  = 1  − 0.657·2-s + 0.765·3-s − 0.567·4-s − 0.503·6-s − 1.63·7-s + 1.03·8-s − 0.413·9-s + 0.266·11-s − 0.434·12-s + 0.743·13-s + 1.07·14-s − 0.110·16-s + 0.0738·17-s + 0.272·18-s − 0.948·19-s − 1.25·21-s − 0.174·22-s − 0.458·23-s + 0.789·24-s − 0.489·26-s − 1.08·27-s + 0.929·28-s + 1.66·29-s + 1.06·31-s − 0.958·32-s + 0.203·33-s − 0.0485·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{6025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ -1)\)

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 \( 1 \)
241 \( 1 - T \)
good2 \( 1 + 0.930T + 2T^{2} \)
3 \( 1 - 1.32T + 3T^{2} \)
7 \( 1 + 4.33T + 7T^{2} \)
11 \( 1 - 0.882T + 11T^{2} \)
13 \( 1 - 2.68T + 13T^{2} \)
17 \( 1 - 0.304T + 17T^{2} \)
19 \( 1 + 4.13T + 19T^{2} \)
23 \( 1 + 2.19T + 23T^{2} \)
29 \( 1 - 8.97T + 29T^{2} \)
31 \( 1 - 5.91T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + 6.07T + 41T^{2} \)
43 \( 1 + 2.71T + 43T^{2} \)
47 \( 1 - 9.50T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 + 7.93T + 59T^{2} \)
61 \( 1 + 1.60T + 61T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 + 9.40T + 71T^{2} \)
73 \( 1 + 4.11T + 73T^{2} \)
79 \( 1 - 9.90T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + 9.48T + 89T^{2} \)
97 \( 1 + 1.41T + 97T^{2} \)
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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

−7.985435302947920395365997772702, −7.16688897751505396406512355264, −6.25276704230041407612903427082, −5.92264133411439635836322850377, −4.55894442057974343943657224927, −3.95272510981277491929141077135, −3.13845647325040403523794602147, −2.46760850790596015197252259901, −1.09155134509130966429656443323, 0, 1.09155134509130966429656443323, 2.46760850790596015197252259901, 3.13845647325040403523794602147, 3.95272510981277491929141077135, 4.55894442057974343943657224927, 5.92264133411439635836322850377, 6.25276704230041407612903427082, 7.16688897751505396406512355264, 7.985435302947920395365997772702

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