Properties

Label 2-2520-840.59-c0-0-4
Degree $2$
Conductor $2520$
Sign $0.292 + 0.956i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.707 − 0.707i)7-s − 0.999i·8-s + (0.866 + 0.499i)10-s + (0.448 + 0.258i)11-s − 1.93i·13-s + (−0.965 − 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)19-s + 0.999·20-s + 0.517·22-s + (1.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (−0.965 − 1.67i)26-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.707 − 0.707i)7-s − 0.999i·8-s + (0.866 + 0.499i)10-s + (0.448 + 0.258i)11-s − 1.93i·13-s + (−0.965 − 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)19-s + 0.999·20-s + 0.517·22-s + (1.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (−0.965 − 1.67i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.047469508\)
\(L(\frac12)\) \(\approx\) \(2.047469508\)
\(L(1)\) 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 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.93iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - 0.517T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.237094599662872022299361411129, −8.028123652595972109561619834189, −7.07783203281241105752953967714, −6.51724706184028021151399160600, −5.87513499041250455681079252683, −4.97150958981125258357750665818, −3.95570015678325692651653741829, −3.12499008856637957405612568825, −2.58130300334677576549847811470, −1.09453377981973160621078244169, 1.75801393952488185007086304113, 2.67526326898411404109116665933, 3.85543577046483735629186634608, 4.56643073284846400636031092293, 5.35175532548822369586174290629, 6.22469949811418642703990589348, 6.61353873060094737500761840612, 7.53854686092347061299452697750, 8.696284024427508379840338053549, 9.105924413298151021794638876012

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