Properties

Label 2-2520-2520.1973-c0-0-2
Degree $2$
Conductor $2520$
Sign $0.329 - 0.944i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
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  + (0.965 − 0.258i)2-s + (−0.382 + 0.923i)3-s + (0.866 − 0.499i)4-s + (0.608 + 0.793i)5-s + (−0.130 + 0.991i)6-s + (0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.793 + 0.608i)10-s + (0.130 + 0.991i)12-s + (−0.478 + 1.78i)13-s + (0.499 + 0.866i)14-s + (−0.965 + 0.258i)15-s + (0.500 − 0.866i)16-s + (−0.866 − 0.500i)18-s − 1.98i·19-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.382 + 0.923i)3-s + (0.866 − 0.499i)4-s + (0.608 + 0.793i)5-s + (−0.130 + 0.991i)6-s + (0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.793 + 0.608i)10-s + (0.130 + 0.991i)12-s + (−0.478 + 1.78i)13-s + (0.499 + 0.866i)14-s + (−0.965 + 0.258i)15-s + (0.500 − 0.866i)16-s + (−0.866 − 0.500i)18-s − 1.98i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.329 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.137576291\)
\(L(\frac12)\) \(\approx\) \(2.137576291\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 + (-0.608 - 0.793i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
good11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.478 - 1.78i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 1.98iT - T^{2} \)
23 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.130 + 0.226i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + 1.93iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.198 + 0.739i)T + (-0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.866 - 0.5i)T^{2} \)
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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

−9.388929239159547080804245964688, −8.883939709972238079529474044690, −7.37881839314507322140368707646, −6.48752259293527406734614496194, −6.17830543346629444845337052056, −5.03888289290041932592788132372, −4.73564904438366696303755019082, −3.64101270564605946473277797248, −2.65632158708743773912322976149, −2.02259646538639197991838586858, 1.14222940338764556488294810146, 2.10649457788998942194840704696, 3.28467621340929554780909670307, 4.31834829799006170069670875922, 5.30027526448504116523607357522, 5.70103728615598723922920033227, 6.43001956068464232946861853303, 7.48806840199182357231001459598, 7.889039737361767139510841151201, 8.475074513340877149620769444900

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