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Properties

Label 2-624-39.32-c1-0-10
Degree $2$
Conductor $624$
Sign $0.888 - 0.459i$
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  + (−0.315 + 1.70i)3-s + (−1.72 − 1.72i)5-s + (0.574 − 2.14i)7-s + (−2.80 − 1.07i)9-s + (1.47 + 5.48i)11-s + (2.76 − 2.31i)13-s + (3.48 − 2.39i)15-s + (1.40 + 2.44i)17-s + (7.85 + 2.10i)19-s + (3.47 + 1.65i)21-s + (1.84 − 3.19i)23-s + 0.959i·25-s + (2.71 − 4.43i)27-s + (1.95 + 1.13i)29-s + (−3.13 + 3.13i)31-s + ⋯
L(s)  = 1  + (−0.182 + 0.983i)3-s + (−0.771 − 0.771i)5-s + (0.217 − 0.810i)7-s + (−0.933 − 0.358i)9-s + (0.443 + 1.65i)11-s + (0.765 − 0.643i)13-s + (0.899 − 0.618i)15-s + (0.341 + 0.592i)17-s + (1.80 + 0.483i)19-s + (0.757 + 0.361i)21-s + (0.384 − 0.666i)23-s + 0.191i·25-s + (0.522 − 0.852i)27-s + (0.363 + 0.209i)29-s + (−0.563 + 0.563i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.888 - 0.459i$
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: \((2,\ 624,\ (\ :1/2),\ 0.888 - 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26583 + 0.307858i\)
\(L(\frac12)\) \(\approx\) \(1.26583 + 0.307858i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.315 - 1.70i)T \)
13 \( 1 + (-2.76 + 2.31i)T \)
good5 \( 1 + (1.72 + 1.72i)T + 5iT^{2} \)
7 \( 1 + (-0.574 + 2.14i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.47 - 5.48i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.40 - 2.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.85 - 2.10i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.84 + 3.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.95 - 1.13i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.13 - 3.13i)T - 31iT^{2} \)
37 \( 1 + (-7.36 + 1.97i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.02 + 1.88i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.416 + 0.240i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.57 - 9.57i)T - 47iT^{2} \)
53 \( 1 + 0.617iT - 53T^{2} \)
59 \( 1 + (1.94 + 0.521i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.667 - 1.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.667 - 2.49i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.81 + 6.76i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.75 + 6.75i)T + 73iT^{2} \)
79 \( 1 - 0.372T + 79T^{2} \)
83 \( 1 + (4.99 + 4.99i)T + 83iT^{2} \)
89 \( 1 + (2.17 + 8.09i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.03 - 0.813i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.59665916638453282287993628534, −9.854278295546100418290583690633, −9.073900111368205996260324879160, −8.035742144046555407728871614268, −7.35814863615787368214407599882, −5.96947461432621730416639800807, −4.80595746837661892056210456011, −4.26874376708489343033501655969, −3.31849364458935048671876136896, −1.09781728006853164314683756643, 1.03799074591458845793612878408, 2.78734080314855418145289139173, 3.55213372201947980137625127874, 5.35210956908941629769628808528, 6.08896000330025621708899966609, 7.03563612496178916292974879111, 7.81544905716036375883965836094, 8.640001808501299242178638659558, 9.470585600606617345453530510631, 11.15292331455998560689256480532

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