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  − 1.45·2-s + 2.41·3-s + 0.110·4-s − 3.50·6-s − 1.25·7-s + 2.74·8-s + 2.82·9-s + 2.46·11-s + 0.266·12-s + 2.06·13-s + 1.81·14-s − 4.20·16-s − 1.42·17-s − 4.09·18-s − 0.479·19-s − 3.01·21-s − 3.58·22-s − 2.87·23-s + 6.62·24-s − 3.00·26-s − 0.433·27-s − 0.137·28-s − 9.26·29-s + 0.180·31-s + 0.623·32-s + 5.94·33-s + 2.06·34-s + ⋯
L(s)  = 1  − 1.02·2-s + 1.39·3-s + 0.0551·4-s − 1.43·6-s − 0.472·7-s + 0.970·8-s + 0.940·9-s + 0.743·11-s + 0.0767·12-s + 0.574·13-s + 0.485·14-s − 1.05·16-s − 0.345·17-s − 0.965·18-s − 0.110·19-s − 0.658·21-s − 0.763·22-s − 0.599·23-s + 1.35·24-s − 0.589·26-s − 0.0833·27-s − 0.0260·28-s − 1.72·29-s + 0.0324·31-s + 0.110·32-s + 1.03·33-s + 0.354·34-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 \( 1 + 1.45T + 2T^{2} \)
3 \( 1 - 2.41T + 3T^{2} \)
7 \( 1 + 1.25T + 7T^{2} \)
11 \( 1 - 2.46T + 11T^{2} \)
13 \( 1 - 2.06T + 13T^{2} \)
17 \( 1 + 1.42T + 17T^{2} \)
19 \( 1 + 0.479T + 19T^{2} \)
23 \( 1 + 2.87T + 23T^{2} \)
29 \( 1 + 9.26T + 29T^{2} \)
31 \( 1 - 0.180T + 31T^{2} \)
37 \( 1 - 0.824T + 37T^{2} \)
41 \( 1 - 1.12T + 41T^{2} \)
43 \( 1 + 3.43T + 43T^{2} \)
47 \( 1 + 3.00T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 - 0.433T + 59T^{2} \)
61 \( 1 + 3.88T + 61T^{2} \)
67 \( 1 - 5.24T + 67T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 + 4.35T + 73T^{2} \)
79 \( 1 + 7.28T + 79T^{2} \)
83 \( 1 + 1.09T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + 9.80T + 97T^{2} \)
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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

−8.129094768925847374486846047280, −7.28971617051867657719777216864, −6.64048857424367422517888232221, −5.73038488357579133937931155540, −4.53474417015480912203847613925, −3.83099581928181860096635190303, −3.22252219246109890428807569663, −2.08506366592910514684656520073, −1.44747044836572282743460766097, 0, 1.44747044836572282743460766097, 2.08506366592910514684656520073, 3.22252219246109890428807569663, 3.83099581928181860096635190303, 4.53474417015480912203847613925, 5.73038488357579133937931155540, 6.64048857424367422517888232221, 7.28971617051867657719777216864, 8.129094768925847374486846047280

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