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Properties

Label 2-624-156.71-c0-0-1
Degree $2$
Conductor $624$
Sign $0.884 - 0.466i$
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  + (0.866 + 0.5i)3-s + (−0.5 − 0.133i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.5i)13-s + (−0.366 + 1.36i)19-s + (−0.366 − 0.366i)21-s − i·25-s + 0.999i·27-s + (−1.36 − 1.36i)31-s + (−1.36 + 0.366i)37-s + 0.999·39-s + (−0.5 − 0.866i)43-s + (−0.633 − 0.366i)49-s + (−1 + 0.999i)57-s + (0.866 + 1.5i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.5 − 0.133i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.5i)13-s + (−0.366 + 1.36i)19-s + (−0.366 − 0.366i)21-s − i·25-s + 0.999i·27-s + (−1.36 − 1.36i)31-s + (−1.36 + 0.366i)37-s + 0.999·39-s + (−0.5 − 0.866i)43-s + (−0.633 − 0.366i)49-s + (−1 + 0.999i)57-s + (0.866 + 1.5i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.884 - 0.466i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :0),\ 0.884 - 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167205413\)
\(L(\frac12)\) \(\approx\) \(1.167205413\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + iT^{2} \)
7 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
37 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T^{2} \)
73 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \)
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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

−10.48360034156459532001290016651, −10.15756792141432078983568493582, −9.092102133078297556782214768338, −8.359478574331013208719861392782, −7.58744331676700049554919834486, −6.40068532569119492022843938523, −5.37697451174760308382154281662, −4.01537162195900884426249633102, −3.39447333876825330398786543911, −1.99021925335528738851688344202, 1.66080852393458879334609530641, 3.00821648968595249902278845203, 3.88269957678740659764596381197, 5.28462216873826760392386694921, 6.63617427160700802228370906291, 7.04529333990062908246421717101, 8.291808360038310585851882589517, 8.982045199344113125128863315988, 9.585682015578541981779421345574, 10.80202412296816236613247243245

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