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Properties

Label 2-1925-385.362-c0-0-0
Degree $2$
Conductor $1925$
Sign $0.913 + 0.406i$
Analytic cond. $0.960700$
Root an. cond. $0.980153$
Motivic weight $0$
Arithmetic yes
Rational no
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 + (−0.707 − 0.707i)7-s + (1.22 + 1.22i)8-s + (0.866 + 0.5i)9-s + (0.5 + 0.866i)11-s + (−0.707 + 0.707i)13-s + (−0.866 + 1.5i)14-s + (0.500 − 0.866i)16-s + (−0.517 + 1.93i)17-s + (0.448 − 1.67i)18-s + (1.22 − 1.22i)22-s + (1.5 + 0.866i)26-s + (1.93 + 0.517i)28-s + (−1.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (−0.448 − 1.67i)2-s + (−1.73 + 1.00i)4-s + (−0.707 − 0.707i)7-s + (1.22 + 1.22i)8-s + (0.866 + 0.5i)9-s + (0.5 + 0.866i)11-s + (−0.707 + 0.707i)13-s + (−0.866 + 1.5i)14-s + (0.500 − 0.866i)16-s + (−0.517 + 1.93i)17-s + (0.448 − 1.67i)18-s + (1.22 − 1.22i)22-s + (1.5 + 0.866i)26-s + (1.93 + 0.517i)28-s + (−1.5 + 0.866i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.913 + 0.406i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.913 + 0.406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6181641734\)
\(L(\frac12)\) \(\approx\) \(0.6181641734\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \)
3 \( 1 + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
17 \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
89 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.501085603083802307169159318100, −9.057280300019180651263868291386, −7.929122506501476132857562165788, −7.13780790590018355690608776244, −6.31906503452416300923476944581, −4.68032784175806648077342614091, −4.16337630395247724794880871266, −3.43999277568273512221755746056, −2.09198253989485913297891970751, −1.50741628151191515651737671680, 0.54674873729983900660566487914, 2.63898555247045160710494623368, 3.86340273407370800647300586240, 4.99476091052162575593440720995, 5.67289087054551008007990953812, 6.38636380248564088786357510020, 7.17710923826551049445281781579, 7.57309841389758698710209534138, 8.721466838787421042024132664430, 9.331034259043737941524362827078

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