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Properties

Label 2-504-7.2-c5-0-17
Degree $2$
Conductor $504$
Sign $-0.654 - 0.756i$
Analytic cond. $80.8334$
Root an. cond. $8.99074$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−45.7 + 79.1i)5-s + (103. − 78.4i)7-s + (361. + 626. i)11-s − 0.466·13-s + (778. + 1.34e3i)17-s + (−1.20e3 + 2.08e3i)19-s + (1.81e3 − 3.14e3i)23-s + (−2.61e3 − 4.53e3i)25-s + 2.49e3·29-s + (157. + 272. i)31-s + (1.48e3 + 1.17e4i)35-s + (2.53e3 − 4.38e3i)37-s + 1.04e4·41-s + 2.40e4·43-s + (−8.98e3 + 1.55e4i)47-s + ⋯
L(s)  = 1  + (−0.817 + 1.41i)5-s + (0.796 − 0.604i)7-s + (0.901 + 1.56i)11-s − 0.000765·13-s + (0.653 + 1.13i)17-s + (−0.766 + 1.32i)19-s + (0.715 − 1.24i)23-s + (−0.837 − 1.45i)25-s + 0.551·29-s + (0.0294 + 0.0509i)31-s + (0.205 + 1.62i)35-s + (0.303 − 0.526i)37-s + 0.968·41-s + 1.98·43-s + (−0.593 + 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.654 - 0.756i$
Analytic conductor: \(80.8334\)
Root analytic conductor: \(8.99074\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :5/2),\ -0.654 - 0.756i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.962594323\)
\(L(\frac12)\) \(\approx\) \(1.962594323\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7 \( 1 + (-103. + 78.4i)T \)
good5 \( 1 + (45.7 - 79.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-361. - 626. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 0.466T + 3.71e5T^{2} \)
17 \( 1 + (-778. - 1.34e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.20e3 - 2.08e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-1.81e3 + 3.14e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 2.49e3T + 2.05e7T^{2} \)
31 \( 1 + (-157. - 272. i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-2.53e3 + 4.38e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.04e4T + 1.15e8T^{2} \)
43 \( 1 - 2.40e4T + 1.47e8T^{2} \)
47 \( 1 + (8.98e3 - 1.55e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.28e3 + 2.23e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.48e4 - 2.57e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-7.92e3 + 1.37e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-4.05e3 - 7.02e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 5.99e4T + 1.80e9T^{2} \)
73 \( 1 + (-9.34e3 - 1.61e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.88e4 - 3.27e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 9.09e4T + 3.93e9T^{2} \)
89 \( 1 + (3.60e4 - 6.24e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 4.90e4T + 8.58e9T^{2} \)
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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

−10.59110003053987081002800754827, −9.847788034817457534120971769949, −8.418777462717578887257738018289, −7.60044771762959519557839827951, −6.97623064637320571945120826831, −6.07401325722936139027063305482, −4.30973316481969430461384937676, −3.98226226789362651152503055098, −2.51650682311416130497233847604, −1.30485781671765514169824585581, 0.53259619081293397636040758668, 1.21765727511211785011781422320, 2.93047659330992302404590667167, 4.18200085900362318870006248189, 5.02883976481582975085994214609, 5.83242768306020584605322056897, 7.26117770154094484020180103868, 8.248668740855010852581510412175, 8.851880409554465911848366855366, 9.369803140638382919366939456394

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