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Properties

Label 2-624-52.31-c3-0-2
Degree $2$
Conductor $624$
Sign $-0.916 + 0.399i$
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  + 3i·3-s + (−2.84 + 2.84i)5-s + (9.35 − 9.35i)7-s − 9·9-s + (−20.3 + 20.3i)11-s + (18.5 − 43.0i)13-s + (−8.53 − 8.53i)15-s + 31.4i·17-s + (32.5 + 32.5i)19-s + (28.0 + 28.0i)21-s − 107.·23-s + 108. i·25-s − 27i·27-s − 231.·29-s + (−3.20 − 3.20i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.254 + 0.254i)5-s + (0.505 − 0.505i)7-s − 0.333·9-s + (−0.557 + 0.557i)11-s + (0.395 − 0.918i)13-s + (−0.146 − 0.146i)15-s + 0.448i·17-s + (0.393 + 0.393i)19-s + (0.291 + 0.291i)21-s − 0.972·23-s + 0.870i·25-s − 0.192i·27-s − 1.48·29-s + (−0.0185 − 0.0185i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.916 + 0.399i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.916 + 0.399i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1463697074\)
\(L(\frac12)\) \(\approx\) \(0.1463697074\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
13 \( 1 + (-18.5 + 43.0i)T \)
good5 \( 1 + (2.84 - 2.84i)T - 125iT^{2} \)
7 \( 1 + (-9.35 + 9.35i)T - 343iT^{2} \)
11 \( 1 + (20.3 - 20.3i)T - 1.33e3iT^{2} \)
17 \( 1 - 31.4iT - 4.91e3T^{2} \)
19 \( 1 + (-32.5 - 32.5i)T + 6.85e3iT^{2} \)
23 \( 1 + 107.T + 1.21e4T^{2} \)
29 \( 1 + 231.T + 2.43e4T^{2} \)
31 \( 1 + (3.20 + 3.20i)T + 2.97e4iT^{2} \)
37 \( 1 + (159. + 159. i)T + 5.06e4iT^{2} \)
41 \( 1 + (68.9 - 68.9i)T - 6.89e4iT^{2} \)
43 \( 1 - 213.T + 7.95e4T^{2} \)
47 \( 1 + (-95.2 + 95.2i)T - 1.03e5iT^{2} \)
53 \( 1 + 647.T + 1.48e5T^{2} \)
59 \( 1 + (-182. + 182. i)T - 2.05e5iT^{2} \)
61 \( 1 + 263.T + 2.26e5T^{2} \)
67 \( 1 + (637. + 637. i)T + 3.00e5iT^{2} \)
71 \( 1 + (-229. - 229. i)T + 3.57e5iT^{2} \)
73 \( 1 + (291. + 291. i)T + 3.89e5iT^{2} \)
79 \( 1 + 633. iT - 4.93e5T^{2} \)
83 \( 1 + (583. + 583. i)T + 5.71e5iT^{2} \)
89 \( 1 + (335. + 335. i)T + 7.04e5iT^{2} \)
97 \( 1 + (955. - 955. i)T - 9.12e5iT^{2} \)
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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

−10.69166041149228505161898961232, −9.994187443344325046073894934120, −9.033533035082500665752149560629, −7.85132521298178111379987937424, −7.50586311697702067328760202728, −6.02322406974307723929280042783, −5.18286151052205339116734856510, −4.08999453115787299318395740177, −3.24008607614061261745469378812, −1.71021098549038634445095940808, 0.04072425759240970652590719258, 1.53803175676026674496879003618, 2.67238974088723301315570005347, 4.04520813744437430156445977840, 5.20050610450338252909000194665, 6.05363630296037609913238841804, 7.11413304444942205573083547022, 8.030966152266213873097158544911, 8.683451796674401882361411617211, 9.557764736597928595602571977581

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