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Properties

Label 2-624-39.2-c1-0-12
Degree $2$
Conductor $624$
Sign $0.280 + 0.959i$
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  + (−1.45 − 0.933i)3-s + (−0.428 − 0.428i)5-s + (0.735 − 0.196i)7-s + (1.25 + 2.72i)9-s + (4.05 + 1.08i)11-s + (0.601 − 3.55i)13-s + (0.224 + 1.02i)15-s + (−2.62 + 4.54i)17-s + (−0.882 − 3.29i)19-s + (−1.25 − 0.399i)21-s + (−0.933 − 1.61i)23-s − 4.63i·25-s + (0.713 − 5.14i)27-s + (7.53 − 4.35i)29-s + (2.68 − 2.68i)31-s + ⋯
L(s)  = 1  + (−0.842 − 0.539i)3-s + (−0.191 − 0.191i)5-s + (0.277 − 0.0744i)7-s + (0.418 + 0.908i)9-s + (1.22 + 0.327i)11-s + (0.166 − 0.986i)13-s + (0.0580 + 0.264i)15-s + (−0.636 + 1.10i)17-s + (−0.202 − 0.755i)19-s + (−0.274 − 0.0871i)21-s + (−0.194 − 0.337i)23-s − 0.926i·25-s + (0.137 − 0.990i)27-s + (1.40 − 0.808i)29-s + (0.481 − 0.481i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.280 + 0.959i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.280 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.882034 - 0.661148i\)
\(L(\frac12)\) \(\approx\) \(0.882034 - 0.661148i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 + 0.933i)T \)
13 \( 1 + (-0.601 + 3.55i)T \)
good5 \( 1 + (0.428 + 0.428i)T + 5iT^{2} \)
7 \( 1 + (-0.735 + 0.196i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.05 - 1.08i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.62 - 4.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.882 + 3.29i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.933 + 1.61i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.53 + 4.35i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.68 + 2.68i)T - 31iT^{2} \)
37 \( 1 + (-1.52 + 5.67i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.29 + 8.56i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.68 + 0.975i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.73 - 5.73i)T - 47iT^{2} \)
53 \( 1 - 9.01iT - 53T^{2} \)
59 \( 1 + (2.23 + 8.34i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.06 + 7.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.101 - 0.0271i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-10.0 + 2.69i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-5.57 - 5.57i)T + 73iT^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + (0.996 + 0.996i)T + 83iT^{2} \)
89 \( 1 + (6.32 + 1.69i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.07 - 15.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.69093655813590541008419975575, −9.676671930001533481143477432203, −8.470713810964949332248682004216, −7.82928802181205177416350907296, −6.59211549650880575746288329704, −6.16027074832229662666877505134, −4.84496219350940440525170753395, −4.05506834133734775815758226616, −2.22801766979318118105000508340, −0.77909095998468494677921778817, 1.36549070182822092667886329154, 3.33035739764419164940915152654, 4.35087733307683689479307756519, 5.16627271009585128322809390706, 6.51313146767066365367204503718, 6.78888837684078811267597912496, 8.306982329040450388713936731050, 9.230033698910577137913555223699, 9.893975751107969920145572679989, 10.96739649130297720111325682228

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