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Properties

Label 2-624-12.11-c3-0-17
Degree $2$
Conductor $624$
Sign $0.438 - 0.898i$
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  + (−2.28 + 4.66i)3-s − 4.73i·5-s − 12.5i·7-s + (−16.5 − 21.2i)9-s − 56.4·11-s − 13·13-s + (22.1 + 10.8i)15-s − 33.1i·17-s + 65.5i·19-s + (58.7 + 28.7i)21-s + 21.2·23-s + 102.·25-s + (137. − 28.9i)27-s + 81.2i·29-s + 235. i·31-s + ⋯
L(s)  = 1  + (−0.438 + 0.898i)3-s − 0.423i·5-s − 0.679i·7-s + (−0.614 − 0.788i)9-s − 1.54·11-s − 0.277·13-s + (0.380 + 0.186i)15-s − 0.472i·17-s + 0.791i·19-s + (0.610 + 0.298i)21-s + 0.192·23-s + 0.820·25-s + (0.978 − 0.206i)27-s + 0.519i·29-s + 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.438 - 0.898i$
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: \((2,\ 624,\ (\ :3/2),\ 0.438 - 0.898i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.113159927\)
\(L(\frac12)\) \(\approx\) \(1.113159927\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.28 - 4.66i)T \)
13 \( 1 + 13T \)
good5 \( 1 + 4.73iT - 125T^{2} \)
7 \( 1 + 12.5iT - 343T^{2} \)
11 \( 1 + 56.4T + 1.33e3T^{2} \)
17 \( 1 + 33.1iT - 4.91e3T^{2} \)
19 \( 1 - 65.5iT - 6.85e3T^{2} \)
23 \( 1 - 21.2T + 1.21e4T^{2} \)
29 \( 1 - 81.2iT - 2.43e4T^{2} \)
31 \( 1 - 235. iT - 2.97e4T^{2} \)
37 \( 1 - 139.T + 5.06e4T^{2} \)
41 \( 1 + 332. iT - 6.89e4T^{2} \)
43 \( 1 - 133. iT - 7.95e4T^{2} \)
47 \( 1 - 286.T + 1.03e5T^{2} \)
53 \( 1 + 52.6iT - 1.48e5T^{2} \)
59 \( 1 - 119.T + 2.05e5T^{2} \)
61 \( 1 + 376.T + 2.26e5T^{2} \)
67 \( 1 - 487. iT - 3.00e5T^{2} \)
71 \( 1 - 466.T + 3.57e5T^{2} \)
73 \( 1 - 461.T + 3.89e5T^{2} \)
79 \( 1 - 533. iT - 4.93e5T^{2} \)
83 \( 1 - 897.T + 5.71e5T^{2} \)
89 \( 1 - 1.51e3iT - 7.04e5T^{2} \)
97 \( 1 - 47.3T + 9.12e5T^{2} \)
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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

−10.62924536093025301149208175420, −9.633721749514420460946703659852, −8.732284604956464658847614116906, −7.77708123156344723391060708800, −6.77551546024372733591819228782, −5.43807289231840892596837892480, −4.97035210570641694975634551096, −3.88046649160226127811006351513, −2.73522682273150901576529958291, −0.807535091052723610056876823516, 0.47652367107410957013746477447, 2.21136224406363993739036547739, 2.86473644978371643545619135072, 4.73202045366806042969929401584, 5.62602895182427558520682880068, 6.41139587271852216912839504904, 7.46543334223600736694025714182, 8.048040080535957866228207525487, 9.082458532932047261523175762525, 10.27538679167062863321988204165

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