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  + 0.630·2-s − 2.33·3-s − 1.60·4-s − 1.47·6-s + 3.68·7-s − 2.27·8-s + 2.46·9-s − 4.96·11-s + 3.74·12-s + 1.69·13-s + 2.32·14-s + 1.77·16-s − 5.52·17-s + 1.55·18-s + 4.21·19-s − 8.60·21-s − 3.13·22-s + 2.77·23-s + 5.31·24-s + 1.06·26-s + 1.24·27-s − 5.90·28-s − 2.31·29-s − 0.199·31-s + 5.66·32-s + 11.6·33-s − 3.48·34-s + ⋯
L(s)  = 1  + 0.445·2-s − 1.34·3-s − 0.801·4-s − 0.601·6-s + 1.39·7-s − 0.803·8-s + 0.822·9-s − 1.49·11-s + 1.08·12-s + 0.468·13-s + 0.620·14-s + 0.442·16-s − 1.33·17-s + 0.366·18-s + 0.966·19-s − 1.87·21-s − 0.667·22-s + 0.578·23-s + 1.08·24-s + 0.209·26-s + 0.240·27-s − 1.11·28-s − 0.429·29-s − 0.0358·31-s + 1.00·32-s + 2.02·33-s − 0.597·34-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 \( 1 - 0.630T + 2T^{2} \)
3 \( 1 + 2.33T + 3T^{2} \)
7 \( 1 - 3.68T + 7T^{2} \)
11 \( 1 + 4.96T + 11T^{2} \)
13 \( 1 - 1.69T + 13T^{2} \)
17 \( 1 + 5.52T + 17T^{2} \)
19 \( 1 - 4.21T + 19T^{2} \)
23 \( 1 - 2.77T + 23T^{2} \)
29 \( 1 + 2.31T + 29T^{2} \)
31 \( 1 + 0.199T + 31T^{2} \)
37 \( 1 + 1.79T + 37T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 - 5.34T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 1.32T + 53T^{2} \)
59 \( 1 - 5.78T + 59T^{2} \)
61 \( 1 + 0.0766T + 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + 5.20T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 - 7.82T + 79T^{2} \)
83 \( 1 + 2.55T + 83T^{2} \)
89 \( 1 + 4.35T + 89T^{2} \)
97 \( 1 + 9.02T + 97T^{2} \)
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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

−7.66732173484191006053910188760, −6.94845908994701804015216219651, −5.89862174202421435185301540945, −5.45600652491650902761008537535, −4.86079972092264136323148936236, −4.54465720128173083169686168159, −3.42874127235488098654666869704, −2.28821761606705352581899603590, −1.04486726593146973910804971440, 0, 1.04486726593146973910804971440, 2.28821761606705352581899603590, 3.42874127235488098654666869704, 4.54465720128173083169686168159, 4.86079972092264136323148936236, 5.45600652491650902761008537535, 5.89862174202421435185301540945, 6.94845908994701804015216219651, 7.66732173484191006053910188760

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