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Properties

Label 2-504-63.4-c1-0-0
Degree $2$
Conductor $504$
Sign $-0.831 - 0.556i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (0.666 + 1.59i)3-s − 0.468·5-s + (−2.39 − 1.13i)7-s + (−2.11 + 2.12i)9-s − 1.34·11-s + (−3.16 + 5.48i)13-s + (−0.311 − 0.748i)15-s + (−2.47 + 4.28i)17-s + (2.38 + 4.13i)19-s + (0.222 − 4.57i)21-s + 7.62·23-s − 4.78·25-s + (−4.81 − 1.95i)27-s + (−1.80 − 3.12i)29-s + (−3.24 − 5.62i)31-s + ⋯
L(s)  = 1  + (0.384 + 0.923i)3-s − 0.209·5-s + (−0.903 − 0.428i)7-s + (−0.704 + 0.709i)9-s − 0.406·11-s + (−0.877 + 1.52i)13-s + (−0.0805 − 0.193i)15-s + (−0.599 + 1.03i)17-s + (0.548 + 0.949i)19-s + (0.0485 − 0.998i)21-s + 1.59·23-s − 0.956·25-s + (−0.926 − 0.377i)27-s + (−0.335 − 0.580i)29-s + (−0.583 − 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.831 - 0.556i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.831 - 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259614 + 0.854598i\)
\(L(\frac12)\) \(\approx\) \(0.259614 + 0.854598i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.666 - 1.59i)T \)
7 \( 1 + (2.39 + 1.13i)T \)
good5 \( 1 + 0.468T + 5T^{2} \)
11 \( 1 + 1.34T + 11T^{2} \)
13 \( 1 + (3.16 - 5.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.47 - 4.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.38 - 4.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.62T + 23T^{2} \)
29 \( 1 + (1.80 + 3.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.24 + 5.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.24 - 9.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0251 - 0.0435i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.431 + 0.748i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.49 + 9.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.84 + 10.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.93 - 3.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.87 - 3.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.32 - 2.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.04T + 71T^{2} \)
73 \( 1 + (3.30 - 5.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.58 - 2.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.90 - 8.49i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.30 - 9.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.97 - 12.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.19563634384176419637905735910, −10.14206535067015832386450725821, −9.651566037486263455706650239049, −8.804327551878322633338225966542, −7.73122326844200839757645731216, −6.76038817545203448192225211410, −5.58338806589948007641365977963, −4.34574272046265913481620108220, −3.66364804165151032914956567720, −2.31550880145490116684722383587, 0.47973028307300541520078393091, 2.60886875618298855547463599865, 3.15914691525975431270270651564, 5.04603922300533493962171097879, 5.94992962873295398596727283690, 7.29711869842743248569836578432, 7.45140332146868664501302679990, 8.928022296430133581397767203012, 9.354078911005655219193108600670, 10.62393157371201042453126607987

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