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Properties

Label 2-624-13.3-c1-0-10
Degree $2$
Conductor $624$
Sign $0.711 + 0.702i$
Analytic cond. $4.98266$
Root an. cond. $2.23218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s − 5-s + (2.30 − 3.98i)7-s + (−0.499 + 0.866i)9-s + (−2.30 − 3.98i)11-s + (1.80 + 3.12i)13-s + (−0.5 − 0.866i)15-s + (2.80 − 4.85i)17-s + (0.302 − 0.524i)19-s + 4.60·21-s + (−2.30 − 3.98i)23-s − 4·25-s − 0.999·27-s + (−1.5 − 2.59i)29-s + 9.21·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 0.447·5-s + (0.870 − 1.50i)7-s + (−0.166 + 0.288i)9-s + (−0.694 − 1.20i)11-s + (0.499 + 0.866i)13-s + (−0.129 − 0.223i)15-s + (0.679 − 1.17i)17-s + (0.0694 − 0.120i)19-s + 1.00·21-s + (−0.480 − 0.831i)23-s − 0.800·25-s − 0.192·27-s + (−0.278 − 0.482i)29-s + 1.65·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42535 - 0.585053i\)
\(L(\frac12)\) \(\approx\) \(1.42535 - 0.585053i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-1.80 - 3.12i)T \)
good5 \( 1 + T + 5T^{2} \)
7 \( 1 + (-2.30 + 3.98i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.30 + 3.98i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.80 + 4.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.302 + 0.524i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.30 + 3.98i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.21T + 31T^{2} \)
37 \( 1 + (-4.80 - 8.31i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.19 - 3.80i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.30 + 3.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.60T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (2.60 - 4.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.197 + 0.341i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.69 + 2.93i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.30 - 14.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 9.81T + 83T^{2} \)
89 \( 1 + (6.21 + 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.60 + 2.78i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.47913275376069960824928412425, −9.829388654175488073862106060193, −8.545093310332128452638729747586, −7.970647309123925437217481836826, −7.17940898645709029504534317659, −5.90267930051870197457653544908, −4.61126447559673643822942090798, −4.04842673792853787467648986722, −2.83208479702693209189775731520, −0.885636292225271726286479775003, 1.70109306669512359647220958390, 2.71800075561963809995043111708, 4.12083469587715386026049211746, 5.42963370913728439275590509487, 5.99424164559264628300038904770, 7.62208591505403314087480665603, 7.924814603179977346097280927811, 8.766869053013918180426670297762, 9.796532055169550743035083262428, 10.77752911621030303810501704475

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