Properties

Label 2-1260-1260.643-c0-0-1
Degree $2$
Conductor $1260$
Sign $0.784 - 0.619i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
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  + (−0.866 + 0.5i)2-s + (−0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + (0.965 + 0.258i)6-s + (−0.965 − 0.258i)7-s + 0.999i·8-s + 1.00i·9-s + (−0.707 − 0.707i)10-s + (0.866 − 0.5i)11-s + (−0.965 + 0.258i)12-s + (−0.258 − 0.965i)13-s + (0.965 − 0.258i)14-s + (0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + (0.965 + 0.258i)6-s + (−0.965 − 0.258i)7-s + 0.999i·8-s + 1.00i·9-s + (−0.707 − 0.707i)10-s + (0.866 − 0.5i)11-s + (−0.965 + 0.258i)12-s + (−0.258 − 0.965i)13-s + (0.965 − 0.258i)14-s + (0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.707 + 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.784 - 0.619i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.784 - 0.619i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5360589805\)
\(L(\frac12)\) \(\approx\) \(0.5360589805\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
good11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
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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

−10.29052042505432216499181585858, −9.225671791322955044230002701626, −8.056858583939199620812741119000, −7.47725896407060140727916834688, −6.67918622897795451968890610879, −5.96373649234806029161281045835, −5.65728314873704411409365513059, −3.74032679380075073628163374079, −2.52940146873883400664021505313, −1.13467938091675404829471435678, 0.830967595032198311776517594613, 2.40700394541610343815090493996, 3.74607989247494386457915952986, 4.51945400730650683252336199682, 5.59926086924605759014766324345, 6.65196359948218367765170621697, 7.21413595436967274331725204506, 8.766964038585687432924747002678, 9.206219254971462568177729916039, 9.640487969231627730219400606310

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