Properties

Label 2-624-39.11-c1-0-14
Degree $2$
Conductor $624$
Sign $0.986 + 0.163i$
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  + (−0.239 + 1.71i)3-s + (1.06 − 1.06i)5-s + (−0.366 − 1.36i)7-s + (−2.88 − 0.820i)9-s + (1.06 − 3.97i)11-s + (3.59 + 0.232i)13-s + (1.57 + 2.08i)15-s + (2.51 − 4.36i)17-s + (3.73 − i)19-s + (2.43 − 0.301i)21-s + 2.73i·25-s + (2.09 − 4.75i)27-s + (−6.20 + 3.58i)29-s + (2.46 + 2.46i)31-s + (6.56 + 2.77i)33-s + ⋯
L(s)  = 1  + (−0.138 + 0.990i)3-s + (0.476 − 0.476i)5-s + (−0.138 − 0.516i)7-s + (−0.961 − 0.273i)9-s + (0.321 − 1.19i)11-s + (0.997 + 0.0643i)13-s + (0.405 + 0.537i)15-s + (0.611 − 1.05i)17-s + (0.856 − 0.229i)19-s + (0.530 − 0.0657i)21-s + 0.546i·25-s + (0.403 − 0.914i)27-s + (−1.15 + 0.665i)29-s + (0.442 + 0.442i)31-s + (1.14 + 0.483i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.986 + 0.163i$
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: \((2,\ 624,\ (\ :1/2),\ 0.986 + 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52949 - 0.125959i\)
\(L(\frac12)\) \(\approx\) \(1.52949 - 0.125959i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.239 - 1.71i)T \)
13 \( 1 + (-3.59 - 0.232i)T \)
good5 \( 1 + (-1.06 + 1.06i)T - 5iT^{2} \)
7 \( 1 + (0.366 + 1.36i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.06 + 3.97i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.51 + 4.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.73 + i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.20 - 3.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.46 - 2.46i)T + 31iT^{2} \)
37 \( 1 + (5.23 + 1.40i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.42 - 1.45i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.90 + 1.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.25 - 4.25i)T + 47iT^{2} \)
53 \( 1 + 0.779iT - 53T^{2} \)
59 \( 1 + (-2.90 + 0.779i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.53 + 5.73i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.779 - 2.90i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.901 - 0.901i)T - 73iT^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + (2.90 - 2.90i)T - 83iT^{2} \)
89 \( 1 + (2.41 - 9.01i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.63 + 0.437i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.67597410869572919082642876347, −9.534956327310378269628503883415, −9.135781390681919862073263218390, −8.202426856638977999080657739374, −6.92759666815694778927253645403, −5.72122261263827855119112108299, −5.22725742127030344698208784566, −3.87435732304216713120584090101, −3.13402229082820806357964382122, −0.990322527641059224456056828872, 1.49650738571084111305570000705, 2.52906036445032653005205020099, 3.89607773803268357450881601478, 5.55122816303582697191429348706, 6.11231154130988006972898389438, 7.03021847301314523588801292371, 7.87912782972042850583559908788, 8.809555355049436007035222801552, 9.809655592383935134690375621540, 10.63162505732510799705912476867

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