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Properties

Label 2-624-39.11-c1-0-7
Degree $2$
Conductor $624$
Sign $0.822 - 0.568i$
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  + (−1.73 + 0.0795i)3-s + (−0.313 + 0.313i)5-s + (0.0745 + 0.278i)7-s + (2.98 − 0.275i)9-s + (0.150 − 0.563i)11-s + (−1.79 − 3.12i)13-s + (0.517 − 0.567i)15-s + (−2.79 + 4.84i)17-s + (6.79 − 1.81i)19-s + (−0.151 − 0.475i)21-s + (3.32 + 5.76i)23-s + 4.80i·25-s + (−5.14 + 0.713i)27-s + (3.57 − 2.06i)29-s + (1.03 + 1.03i)31-s + ⋯
L(s)  = 1  + (−0.998 + 0.0459i)3-s + (−0.140 + 0.140i)5-s + (0.0281 + 0.105i)7-s + (0.995 − 0.0917i)9-s + (0.0454 − 0.169i)11-s + (−0.496 − 0.867i)13-s + (0.133 − 0.146i)15-s + (−0.678 + 1.17i)17-s + (1.55 − 0.417i)19-s + (−0.0329 − 0.103i)21-s + (0.693 + 1.20i)23-s + 0.960i·25-s + (−0.990 + 0.137i)27-s + (0.664 − 0.383i)29-s + (0.186 + 0.186i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.822 - 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966350 + 0.301325i\)
\(L(\frac12)\) \(\approx\) \(0.966350 + 0.301325i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0795i)T \)
13 \( 1 + (1.79 + 3.12i)T \)
good5 \( 1 + (0.313 - 0.313i)T - 5iT^{2} \)
7 \( 1 + (-0.0745 - 0.278i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.150 + 0.563i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.79 - 4.84i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.79 + 1.81i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.32 - 5.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.57 + 2.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.03 - 1.03i)T + 31iT^{2} \)
37 \( 1 + (-6.72 - 1.80i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-7.36 - 1.97i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.26 - 1.88i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.71 - 3.71i)T + 47iT^{2} \)
53 \( 1 + 3.64iT - 53T^{2} \)
59 \( 1 + (3.29 - 0.881i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.25 - 9.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.29 - 8.55i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.98 + 14.8i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.52 + 3.52i)T - 73iT^{2} \)
79 \( 1 + 1.10T + 79T^{2} \)
83 \( 1 + (8.23 - 8.23i)T - 83iT^{2} \)
89 \( 1 + (3.64 - 13.6i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.59 + 0.694i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.87042920316927404284226755008, −9.926048860435383459979961300602, −9.156627906802507897198621390799, −7.80369711918104262103059096255, −7.16967479655136824448640356631, −6.00623558733980350430725899315, −5.34561364098185792405380759311, −4.27626061408199368904151714948, −2.98586266344475488677684892330, −1.13443936615048352088504231463, 0.804558355952109496583874236877, 2.54754776833843621201789562614, 4.29656047015356637675976648098, 4.88182657381645288152088366848, 6.01707836930933873220183905503, 6.97088810617152334494360944318, 7.58093722116243738280975195770, 8.984572069656447184437255508486, 9.714604654363428868563313236949, 10.63409551672202807248775934679

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