Properties

Label 2-624-39.32-c1-0-13
Degree $2$
Conductor $624$
Sign $0.815 - 0.579i$
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  + (1.43 − 0.968i)3-s + (2.19 + 2.19i)5-s + (−1.17 + 4.39i)7-s + (1.12 − 2.78i)9-s + (−0.917 − 3.42i)11-s + (0.225 + 3.59i)13-s + (5.28 + 1.02i)15-s + (2.20 + 3.81i)17-s + (−1.06 − 0.284i)19-s + (2.56 + 7.45i)21-s + (0.812 − 1.40i)23-s + 4.67i·25-s + (−1.08 − 5.08i)27-s + (−4.61 − 2.66i)29-s + (−3.28 + 3.28i)31-s + ⋯
L(s)  = 1  + (0.828 − 0.559i)3-s + (0.983 + 0.983i)5-s + (−0.445 + 1.66i)7-s + (0.374 − 0.927i)9-s + (−0.276 − 1.03i)11-s + (0.0624 + 0.998i)13-s + (1.36 + 0.265i)15-s + (0.534 + 0.926i)17-s + (−0.243 − 0.0653i)19-s + (0.560 + 1.62i)21-s + (0.169 − 0.293i)23-s + 0.935i·25-s + (−0.208 − 0.977i)27-s + (−0.857 − 0.494i)29-s + (−0.589 + 0.589i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.815 - 0.579i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.815 - 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05463 + 0.655374i\)
\(L(\frac12)\) \(\approx\) \(2.05463 + 0.655374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 + 0.968i)T \)
13 \( 1 + (-0.225 - 3.59i)T \)
good5 \( 1 + (-2.19 - 2.19i)T + 5iT^{2} \)
7 \( 1 + (1.17 - 4.39i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.917 + 3.42i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.20 - 3.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.06 + 0.284i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.812 + 1.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.61 + 2.66i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.28 - 3.28i)T - 31iT^{2} \)
37 \( 1 + (-2.75 + 0.737i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-9.76 + 2.61i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.07 + 3.50i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.14 + 4.14i)T - 47iT^{2} \)
53 \( 1 + 8.33iT - 53T^{2} \)
59 \( 1 + (-1.97 - 0.529i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.77 + 8.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.42 - 5.33i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.43 + 5.34i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.99 - 2.99i)T + 73iT^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + (-6.38 - 6.38i)T + 83iT^{2} \)
89 \( 1 + (1.98 + 7.39i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.75 + 2.34i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.62937570567063628626862527844, −9.547199753849685311866952820993, −9.020217793714922541301818591631, −8.246598243492828897905852585430, −7.01764101946360147595096842230, −6.10398694700813526785215514857, −5.73042982011754002341430587547, −3.66937842426649636302184585429, −2.64098623377673383044127777593, −2.00227565838619458835589973378, 1.19826266536371883229228467428, 2.71237650861000535568754758550, 4.00200194248535634686997270477, 4.79909805179038490314363184982, 5.78250733934091992407030943676, 7.39985776916481853590560599274, 7.69044478343267277967777060235, 9.141524108837117974020806353055, 9.612702944327704316059782858638, 10.25119185061128497594198651553

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