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  + 1.88·2-s + 2.55·3-s + 1.54·4-s + 4.81·6-s − 3.51·7-s − 0.856·8-s + 3.53·9-s − 1.11·11-s + 3.95·12-s − 0.230·13-s − 6.62·14-s − 4.70·16-s − 4.35·17-s + 6.65·18-s + 3.14·19-s − 8.99·21-s − 2.10·22-s − 6.36·23-s − 2.18·24-s − 0.434·26-s + 1.37·27-s − 5.43·28-s − 0.0527·29-s − 4.55·31-s − 7.14·32-s − 2.85·33-s − 8.19·34-s + ⋯
L(s)  = 1  + 1.33·2-s + 1.47·3-s + 0.772·4-s + 1.96·6-s − 1.32·7-s − 0.302·8-s + 1.17·9-s − 0.336·11-s + 1.14·12-s − 0.0639·13-s − 1.77·14-s − 1.17·16-s − 1.05·17-s + 1.56·18-s + 0.721·19-s − 1.96·21-s − 0.448·22-s − 1.32·23-s − 0.446·24-s − 0.0851·26-s + 0.264·27-s − 1.02·28-s − 0.00978·29-s − 0.818·31-s − 1.26·32-s − 0.497·33-s − 1.40·34-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 \( 1 - 1.88T + 2T^{2} \)
3 \( 1 - 2.55T + 3T^{2} \)
7 \( 1 + 3.51T + 7T^{2} \)
11 \( 1 + 1.11T + 11T^{2} \)
13 \( 1 + 0.230T + 13T^{2} \)
17 \( 1 + 4.35T + 17T^{2} \)
19 \( 1 - 3.14T + 19T^{2} \)
23 \( 1 + 6.36T + 23T^{2} \)
29 \( 1 + 0.0527T + 29T^{2} \)
31 \( 1 + 4.55T + 31T^{2} \)
37 \( 1 + 6.27T + 37T^{2} \)
41 \( 1 + 8.79T + 41T^{2} \)
43 \( 1 - 2.95T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 - 6.03T + 53T^{2} \)
59 \( 1 - 6.43T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 2.08T + 67T^{2} \)
71 \( 1 + 5.17T + 71T^{2} \)
73 \( 1 - 1.25T + 73T^{2} \)
79 \( 1 + 0.749T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + 8.15T + 89T^{2} \)
97 \( 1 + 7.78T + 97T^{2} \)
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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

−7.52529409982824285760935917116, −7.00191130336933696692378216539, −6.19191354238475288668406098664, −5.55068665098005815961196982938, −4.59005638140708200017710398022, −3.73822773514203012928963746227, −3.44853511261350820762954807577, −2.63209354524789554353983359671, −2.03555694738753461013539067458, 0, 2.03555694738753461013539067458, 2.63209354524789554353983359671, 3.44853511261350820762954807577, 3.73822773514203012928963746227, 4.59005638140708200017710398022, 5.55068665098005815961196982938, 6.19191354238475288668406098664, 7.00191130336933696692378216539, 7.52529409982824285760935917116

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