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Properties

Label 2-624-624.389-c0-0-2
Degree $2$
Conductor $624$
Sign $0.923 + 0.382i$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 + 0.923i)2-s + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)4-s + (−0.541 − 0.541i)5-s + (−0.382 − 0.923i)6-s + (0.923 − 0.382i)8-s − 1.00i·9-s + (0.707 − 0.292i)10-s + (−1.30 − 1.30i)11-s + 12-s + (0.707 − 0.707i)13-s + 0.765·15-s + i·16-s + (0.923 + 0.382i)18-s + 0.765i·20-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)4-s + (−0.541 − 0.541i)5-s + (−0.382 − 0.923i)6-s + (0.923 − 0.382i)8-s − 1.00i·9-s + (0.707 − 0.292i)10-s + (−1.30 − 1.30i)11-s + 12-s + (0.707 − 0.707i)13-s + 0.765·15-s + i·16-s + (0.923 + 0.382i)18-s + 0.765i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3885552101\)
\(L(\frac12)\) \(\approx\) \(0.3885552101\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 0.765iT - T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 - 1.84T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.84iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
89 \( 1 + 1.84iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.62996267143724590448146306485, −9.970534337350792000519509291224, −8.663351227624629711193134610752, −8.398239953341882053270811717965, −7.27220233756671052187405079708, −5.97212492114191607341116343160, −5.51025140663903375047104752966, −4.54734357132608278599234076129, −3.42137822706251170721136390781, −0.56799868889114961379323554390, 1.70273815481864581087862194144, 2.85007558694573951370941669280, 4.27922511120601303739164792078, 5.20494614819500860783729876604, 6.63351974310866636172805478744, 7.53189481950175268137597954106, 8.053567689489737256518928589955, 9.331087558025482653565714219753, 10.34555224713936121295676255561, 10.91248546281821422362868763912

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