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Properties

Label 2-624-12.11-c3-0-16
Degree $2$
Conductor $624$
Sign $0.909 - 0.416i$
Analytic cond. $36.8171$
Root an. cond. $6.06771$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.486 − 5.17i)3-s − 4.15i·5-s + 6.70i·7-s + (−26.5 − 5.03i)9-s − 32.5·11-s − 13·13-s + (−21.4 − 2.01i)15-s + 99.7i·17-s + 73.6i·19-s + (34.6 + 3.26i)21-s + 193.·23-s + 107.·25-s + (−38.9 + 134. i)27-s − 138. i·29-s + 165. i·31-s + ⋯
L(s)  = 1  + (0.0935 − 0.995i)3-s − 0.371i·5-s + 0.362i·7-s + (−0.982 − 0.186i)9-s − 0.893·11-s − 0.277·13-s + (−0.369 − 0.0347i)15-s + 1.42i·17-s + 0.889i·19-s + (0.360 + 0.0338i)21-s + 1.75·23-s + 0.862·25-s + (−0.277 + 0.960i)27-s − 0.889i·29-s + 0.957i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.909 - 0.416i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ 0.909 - 0.416i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.411885530\)
\(L(\frac12)\) \(\approx\) \(1.411885530\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.486 + 5.17i)T \)
13 \( 1 + 13T \)
good5 \( 1 + 4.15iT - 125T^{2} \)
7 \( 1 - 6.70iT - 343T^{2} \)
11 \( 1 + 32.5T + 1.33e3T^{2} \)
17 \( 1 - 99.7iT - 4.91e3T^{2} \)
19 \( 1 - 73.6iT - 6.85e3T^{2} \)
23 \( 1 - 193.T + 1.21e4T^{2} \)
29 \( 1 + 138. iT - 2.43e4T^{2} \)
31 \( 1 - 165. iT - 2.97e4T^{2} \)
37 \( 1 + 54.1T + 5.06e4T^{2} \)
41 \( 1 + 512. iT - 6.89e4T^{2} \)
43 \( 1 - 336. iT - 7.95e4T^{2} \)
47 \( 1 + 495.T + 1.03e5T^{2} \)
53 \( 1 - 537. iT - 1.48e5T^{2} \)
59 \( 1 + 315.T + 2.05e5T^{2} \)
61 \( 1 + 133.T + 2.26e5T^{2} \)
67 \( 1 - 131. iT - 3.00e5T^{2} \)
71 \( 1 - 1.07e3T + 3.57e5T^{2} \)
73 \( 1 - 1.02e3T + 3.89e5T^{2} \)
79 \( 1 - 108. iT - 4.93e5T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 + 139. iT - 7.04e5T^{2} \)
97 \( 1 - 1.45e3T + 9.12e5T^{2} \)
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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

−10.41553928887163934092233808257, −9.135967179697508033463962547981, −8.417218269743309349431996491444, −7.71402939385298672596555963616, −6.72412858097908995004410037341, −5.78344678793859275376861792055, −4.93400359421202921385106579357, −3.36544401159666962613779680871, −2.24946815591710717531989634489, −1.07612750974065370146660709953, 0.45844214550100682875409660122, 2.63847775552633544802312284144, 3.29703764760023673497753506706, 4.84859922273848467203334478062, 5.10385665801929920544242217006, 6.66445244762897687322123461343, 7.45565880761019895063271887839, 8.578479508465240680058562267714, 9.415272295797692630889085594534, 10.10982607979608139227849868974

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