Properties

Degree $2$
Conductor $6025$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 0.185·2-s + 0.636·3-s − 1.96·4-s + 0.118·6-s − 3.56·7-s − 0.735·8-s − 2.59·9-s + 1.87·11-s − 1.25·12-s − 2.26·13-s − 0.661·14-s + 3.79·16-s + 7.09·17-s − 0.481·18-s + 3.44·19-s − 2.27·21-s + 0.347·22-s + 2.84·23-s − 0.467·24-s − 0.420·26-s − 3.56·27-s + 7.01·28-s + 1.58·29-s − 3.99·31-s + 2.17·32-s + 1.19·33-s + 1.31·34-s + ⋯
L(s)  = 1  + 0.131·2-s + 0.367·3-s − 0.982·4-s + 0.0481·6-s − 1.34·7-s − 0.259·8-s − 0.864·9-s + 0.564·11-s − 0.361·12-s − 0.628·13-s − 0.176·14-s + 0.948·16-s + 1.72·17-s − 0.113·18-s + 0.790·19-s − 0.495·21-s + 0.0740·22-s + 0.592·23-s − 0.0955·24-s − 0.0824·26-s − 0.685·27-s + 1.32·28-s + 0.294·29-s − 0.717·31-s + 0.384·32-s + 0.207·33-s + 0.225·34-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 \( 1 - 0.185T + 2T^{2} \)
3 \( 1 - 0.636T + 3T^{2} \)
7 \( 1 + 3.56T + 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 + 2.26T + 13T^{2} \)
17 \( 1 - 7.09T + 17T^{2} \)
19 \( 1 - 3.44T + 19T^{2} \)
23 \( 1 - 2.84T + 23T^{2} \)
29 \( 1 - 1.58T + 29T^{2} \)
31 \( 1 + 3.99T + 31T^{2} \)
37 \( 1 + 1.44T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 8.31T + 43T^{2} \)
47 \( 1 - 1.88T + 47T^{2} \)
53 \( 1 + 7.12T + 53T^{2} \)
59 \( 1 - 7.62T + 59T^{2} \)
61 \( 1 + 5.88T + 61T^{2} \)
67 \( 1 + 0.787T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 - 1.01T + 73T^{2} \)
79 \( 1 - 8.92T + 79T^{2} \)
83 \( 1 - 5.72T + 83T^{2} \)
89 \( 1 + 4.93T + 89T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.80297942607222132996603867303, −7.06947557066313346364358348607, −6.18401538089799430551976590369, −5.50092296501064346400546881792, −4.97113137774025795321644640120, −3.68026909473028322174454789151, −3.42689633985788338451450302968, −2.66831666314136672977971833489, −1.11004918837655028946556811231, 0, 1.11004918837655028946556811231, 2.66831666314136672977971833489, 3.42689633985788338451450302968, 3.68026909473028322174454789151, 4.97113137774025795321644640120, 5.50092296501064346400546881792, 6.18401538089799430551976590369, 7.06947557066313346364358348607, 7.80297942607222132996603867303

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