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Properties

Label 2-624-13.3-c3-0-1
Degree $2$
Conductor $624$
Sign $-0.998 - 0.0583i$
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  + (−1.5 − 2.59i)3-s + 6.42·5-s + (−14.7 + 25.5i)7-s + (−4.5 + 7.79i)9-s + (−0.312 − 0.541i)11-s + (44.3 + 15.0i)13-s + (−9.63 − 16.6i)15-s + (−43.8 + 75.9i)17-s + (41.4 − 71.7i)19-s + 88.4·21-s + (−37.3 − 64.7i)23-s − 83.7·25-s + 27·27-s + (−113. − 196. i)29-s − 173.·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 0.574·5-s + (−0.796 + 1.37i)7-s + (−0.166 + 0.288i)9-s + (−0.00856 − 0.0148i)11-s + (0.947 + 0.320i)13-s + (−0.165 − 0.287i)15-s + (−0.625 + 1.08i)17-s + (0.499 − 0.865i)19-s + 0.919·21-s + (−0.339 − 0.587i)23-s − 0.670·25-s + 0.192·27-s + (−0.724 − 1.25i)29-s − 1.00·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0583i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.998 - 0.0583i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.998 - 0.0583i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1568640800\)
\(L(\frac12)\) \(\approx\) \(0.1568640800\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 + 2.59i)T \)
13 \( 1 + (-44.3 - 15.0i)T \)
good5 \( 1 - 6.42T + 125T^{2} \)
7 \( 1 + (14.7 - 25.5i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (0.312 + 0.541i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (43.8 - 75.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-41.4 + 71.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (37.3 + 64.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (113. + 196. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 173.T + 2.97e4T^{2} \)
37 \( 1 + (56.0 + 97.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-133. - 231. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-191. + 332. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 337.T + 1.03e5T^{2} \)
53 \( 1 + 146.T + 1.48e5T^{2} \)
59 \( 1 + (264. - 458. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (101. - 176. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-60.7 - 105. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (330. - 572. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 167.T + 3.89e5T^{2} \)
79 \( 1 - 101.T + 4.93e5T^{2} \)
83 \( 1 - 506.T + 5.71e5T^{2} \)
89 \( 1 + (701. + 1.21e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (951. - 1.64e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.73481083434187079416322075776, −9.562186147536340643346178733487, −8.996517777878308169479620727607, −8.122113921732464389914145590307, −6.81945914323629931970677450121, −6.00729111019629175605110286890, −5.61842848075686834813286446158, −4.04200261651840635055887491749, −2.64776392577271267467931909954, −1.74982928580601839459824547853, 0.04560375974541999764423656593, 1.43761757133638017364565335710, 3.26475221085730458736898093705, 3.95463641498309529153793364098, 5.21777918259735351198851966660, 6.12868576359012122136917081242, 7.00936003698757947174903043831, 7.921839326150254079177783292333, 9.370361808664753012031063019793, 9.657574087225006819563055174661

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