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  − 1.09·2-s − 0.223·3-s − 0.800·4-s + 0.245·6-s − 2.26·7-s + 3.06·8-s − 2.94·9-s − 3.16·11-s + 0.179·12-s + 0.0649·13-s + 2.47·14-s − 1.75·16-s − 4.44·17-s + 3.23·18-s + 7.82·19-s + 0.506·21-s + 3.46·22-s + 2.53·23-s − 0.686·24-s − 0.0711·26-s + 1.33·27-s + 1.81·28-s − 6.53·29-s + 8.27·31-s − 4.20·32-s + 0.709·33-s + 4.86·34-s + ⋯
L(s)  = 1  − 0.774·2-s − 0.129·3-s − 0.400·4-s + 0.100·6-s − 0.855·7-s + 1.08·8-s − 0.983·9-s − 0.954·11-s + 0.0517·12-s + 0.0180·13-s + 0.662·14-s − 0.439·16-s − 1.07·17-s + 0.761·18-s + 1.79·19-s + 0.110·21-s + 0.739·22-s + 0.527·23-s − 0.140·24-s − 0.0139·26-s + 0.256·27-s + 0.342·28-s − 1.21·29-s + 1.48·31-s − 0.744·32-s + 0.123·33-s + 0.834·34-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 \( 1 + 1.09T + 2T^{2} \)
3 \( 1 + 0.223T + 3T^{2} \)
7 \( 1 + 2.26T + 7T^{2} \)
11 \( 1 + 3.16T + 11T^{2} \)
13 \( 1 - 0.0649T + 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 - 7.82T + 19T^{2} \)
23 \( 1 - 2.53T + 23T^{2} \)
29 \( 1 + 6.53T + 29T^{2} \)
31 \( 1 - 8.27T + 31T^{2} \)
37 \( 1 - 2.73T + 37T^{2} \)
41 \( 1 + 7.52T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + 8.53T + 47T^{2} \)
53 \( 1 + 0.0654T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 7.88T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 + 8.56T + 71T^{2} \)
73 \( 1 - 16.6T + 73T^{2} \)
79 \( 1 + 0.725T + 79T^{2} \)
83 \( 1 - 6.57T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 9.64T + 97T^{2} \)
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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

−7.969399745707588987742254563605, −7.11141948018389680980895998727, −6.45736490346984409662596834750, −5.38355271356636235257423216862, −5.11587821034922205750368166174, −3.97434678570362253451772658224, −3.10640564100693777130520441893, −2.34390942653616083823419031935, −0.908318805671807676462270208885, 0, 0.908318805671807676462270208885, 2.34390942653616083823419031935, 3.10640564100693777130520441893, 3.97434678570362253451772658224, 5.11587821034922205750368166174, 5.38355271356636235257423216862, 6.45736490346984409662596834750, 7.11141948018389680980895998727, 7.969399745707588987742254563605

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