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Properties

Label 2-624-39.20-c1-0-5
Degree $2$
Conductor $624$
Sign $-0.621 - 0.783i$
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  + (−0.0795 + 1.73i)3-s + (0.428 − 0.428i)5-s + (0.735 + 0.196i)7-s + (−2.98 − 0.275i)9-s + (−4.05 + 1.08i)11-s + (0.601 + 3.55i)13-s + (0.707 + 0.775i)15-s + (2.62 + 4.54i)17-s + (−0.882 + 3.29i)19-s + (−0.399 + 1.25i)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  + (−0.0459 + 0.998i)3-s + (0.191 − 0.191i)5-s + (0.277 + 0.0744i)7-s + (−0.995 − 0.0917i)9-s + (−1.22 + 0.327i)11-s + (0.166 + 0.986i)13-s + (0.182 + 0.200i)15-s + (0.636 + 1.10i)17-s + (−0.202 + 0.755i)19-s + (−0.0871 + 0.274i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.621 - 0.783i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498023 + 1.03122i\)
\(L(\frac12)\) \(\approx\) \(0.498023 + 1.03122i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0795 - 1.73i)T \)
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

−10.72536714293484877558790371360, −10.12847123918406029122351200454, −9.279858622282434122939662012426, −8.415471701823455551051622785611, −7.59138309729345964627041439057, −6.14602176726042366626405248625, −5.36099472279673024691348646810, −4.43872840066948219905456342240, −3.43018519907425588519199808283, −1.97209615854109791182815720512, 0.61391056320389551695068273760, 2.32879860020149250087190492593, 3.21261525166155629787519539536, 5.08423334454239478680012074475, 5.69130695482744329351543948949, 6.84534164051405016459992671968, 7.70534979617792039778217921171, 8.243345145821131706606383070789, 9.386518381845101987220683206399, 10.48162541787513724544502825708

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