Properties

Label 2-624-624.77-c0-0-3
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  + (−0.923 + 0.382i)2-s + (0.707 + 0.707i)3-s + (0.707 − 0.707i)4-s + (1.30 − 1.30i)5-s + (−0.923 − 0.382i)6-s + (−0.382 + 0.923i)8-s + 1.00i·9-s + (−0.707 + 1.70i)10-s + (−0.541 + 0.541i)11-s + 12-s + (−0.707 − 0.707i)13-s + 1.84·15-s i·16-s + (−0.382 − 0.923i)18-s − 1.84i·20-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)2-s + (0.707 + 0.707i)3-s + (0.707 − 0.707i)4-s + (1.30 − 1.30i)5-s + (−0.923 − 0.382i)6-s + (−0.382 + 0.923i)8-s + 1.00i·9-s + (−0.707 + 1.70i)10-s + (−0.541 + 0.541i)11-s + 12-s + (−0.707 − 0.707i)13-s + 1.84·15-s i·16-s + (−0.382 − 0.923i)18-s − 1.84i·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: $\chi_{624} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8961996099\)
\(L(\frac12)\) \(\approx\) \(0.8961996099\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.16251444993228925113357358920, −9.913169754619372803195459801714, −9.266865605334552413384404666163, −8.355874918583961019319674442757, −7.81497252812353444474036567915, −6.39280428030400885553877182178, −5.22209901798347140618873888210, −4.83691777022271644997221636205, −2.74691665918881862604545528709, −1.68807888608735225006993147654, 1.87201397448150074188789799761, 2.56558107265464721939571014085, 3.49626980091899211900167495765, 5.70882220329959930277198826926, 6.76133677002433702921005084271, 7.12937920842458185781815718334, 8.228766360164405810188773788314, 9.166082201749211922984570747704, 9.849293193517854592864350317746, 10.53922636886570934371481576867

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