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Properties

Label 2-1260-1260.643-c0-0-0
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  + (−0.5 − 0.866i)2-s + (−0.707 − 0.707i)3-s + (−0.499 + 0.866i)4-s + (−0.258 − 0.965i)5-s + (−0.258 + 0.965i)6-s + (−0.965 − 0.258i)7-s + 0.999·8-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 + (0.258 + 0.965i)14-s + (−0.500 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.707 − 0.707i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.707 − 0.707i)3-s + (−0.499 + 0.866i)4-s + (−0.258 − 0.965i)5-s + (−0.258 + 0.965i)6-s + (−0.965 − 0.258i)7-s + 0.999·8-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 + (0.258 + 0.965i)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

\(L(\frac{1}{2})\) \(\approx\) \(0.1624505676\)
\(L(\frac12)\) \(\approx\) \(0.1624505676\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
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

−10.03917375717574547706833039787, −9.260336588053680347060852244084, −8.434818982429577261988082960410, −7.58406499049621960556002750981, −6.91141907078004746229230815986, −5.74746185732176705886632246841, −4.72399485465396908515286881164, −3.91498746518809432027181761464, −2.47909056263184480900536016581, −1.36653042965177314555976776395, 0.18737140778817125377059364956, 2.77688811522051189595600002347, 3.78631325972753904022059746087, 4.98971840061662865893144092170, 5.82906377687909698924198057211, 6.45651873690275570782728486654, 7.12783076252780754221279975980, 8.209774990372482785658539717299, 9.005880714344387332215923688746, 9.865285793930235054401279603003

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