Properties

Degree $2$
Conductor $1225$
Sign $-0.967 - 0.254i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.448 + 1.67i)2-s + (−1.73 + 1.00i)4-s + (−1.22 − 1.22i)8-s + (0.866 + 0.5i)9-s + (0.5 + 0.866i)11-s + (0.500 − 0.866i)16-s + (−0.448 + 1.67i)18-s + (−1.22 + 1.22i)22-s + (−1.67 + 0.448i)23-s i·29-s − 2·36-s + (−0.448 − 1.67i)37-s + (1.22 + 1.22i)43-s + (−1.73 − 0.999i)44-s + (−1.50 − 2.59i)46-s + ⋯
L(s)  = 1  + (0.448 + 1.67i)2-s + (−1.73 + 1.00i)4-s + (−1.22 − 1.22i)8-s + (0.866 + 0.5i)9-s + (0.5 + 0.866i)11-s + (0.500 − 0.866i)16-s + (−0.448 + 1.67i)18-s + (−1.22 + 1.22i)22-s + (−1.67 + 0.448i)23-s i·29-s − 2·36-s + (−0.448 − 1.67i)37-s + (1.22 + 1.22i)43-s + (−1.73 − 0.999i)44-s + (−1.50 − 2.59i)46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.967 - 0.254i$
Motivic weight: \(0\)
Character: $\chi_{1225} (618, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :0),\ -0.967 - 0.254i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.253404666\)
\(L(\frac12)\) \(\approx\) \(1.253404666\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
3 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.898463940325700867872528587196, −9.387283160420813157629406358735, −8.259569190457153693089973165668, −7.62828153376467313284627957721, −7.04611701723352898002165096824, −6.18397927557874651690128700270, −5.41763965168010985641671744212, −4.36941610548638409971685415239, −3.96349469404813819988081851981, −2.02665743597294185186957471526, 1.06208500442160328966174418076, 2.17338782806571675765020096539, 3.41650230896378843816209700367, 3.96647107914123006314442429121, 4.90769542363948010905261956956, 5.99340255110725250179377493561, 6.94864420413715345271757100080, 8.258017116831273025968414835014, 9.070432743677671488912801103881, 9.857573619183519056394542995083

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