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  − 2.67·2-s − 0.199·3-s + 5.16·4-s + 0.533·6-s + 2.87·7-s − 8.45·8-s − 2.96·9-s − 0.806·11-s − 1.02·12-s + 5.09·13-s − 7.69·14-s + 12.3·16-s + 0.390·17-s + 7.92·18-s + 0.489·19-s − 0.572·21-s + 2.15·22-s + 1.83·23-s + 1.68·24-s − 13.6·26-s + 1.18·27-s + 14.8·28-s − 8.05·29-s − 5.55·31-s − 16.0·32-s + 0.160·33-s − 1.04·34-s + ⋯
L(s)  = 1  − 1.89·2-s − 0.115·3-s + 2.58·4-s + 0.217·6-s + 1.08·7-s − 2.99·8-s − 0.986·9-s − 0.243·11-s − 0.296·12-s + 1.41·13-s − 2.05·14-s + 3.07·16-s + 0.0947·17-s + 1.86·18-s + 0.112·19-s − 0.124·21-s + 0.460·22-s + 0.383·23-s + 0.344·24-s − 2.67·26-s + 0.228·27-s + 2.80·28-s − 1.49·29-s − 0.997·31-s − 2.83·32-s + 0.0279·33-s − 0.179·34-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 \( 1 + 2.67T + 2T^{2} \)
3 \( 1 + 0.199T + 3T^{2} \)
7 \( 1 - 2.87T + 7T^{2} \)
11 \( 1 + 0.806T + 11T^{2} \)
13 \( 1 - 5.09T + 13T^{2} \)
17 \( 1 - 0.390T + 17T^{2} \)
19 \( 1 - 0.489T + 19T^{2} \)
23 \( 1 - 1.83T + 23T^{2} \)
29 \( 1 + 8.05T + 29T^{2} \)
31 \( 1 + 5.55T + 31T^{2} \)
37 \( 1 - 2.20T + 37T^{2} \)
41 \( 1 + 3.59T + 41T^{2} \)
43 \( 1 - 9.27T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 1.68T + 53T^{2} \)
59 \( 1 + 5.02T + 59T^{2} \)
61 \( 1 - 3.39T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 - 8.08T + 79T^{2} \)
83 \( 1 + 0.366T + 83T^{2} \)
89 \( 1 + 2.24T + 89T^{2} \)
97 \( 1 + 7.32T + 97T^{2} \)
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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

−7.83880351650686603439337513816, −7.46772471115871267205165395999, −6.45412456810678414413824019932, −5.85514290239501919200343652114, −5.15237348485785276806269543749, −3.75662561635053973693714046432, −2.86041960070644048704194654200, −1.87628971550945880663218383967, −1.21180458376040085373590439801, 0, 1.21180458376040085373590439801, 1.87628971550945880663218383967, 2.86041960070644048704194654200, 3.75662561635053973693714046432, 5.15237348485785276806269543749, 5.85514290239501919200343652114, 6.45412456810678414413824019932, 7.46772471115871267205165395999, 7.83880351650686603439337513816

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