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Properties

Label 2-504-21.5-c3-0-18
Degree $2$
Conductor $504$
Sign $-0.779 + 0.626i$
Analytic cond. $29.7369$
Root an. cond. $5.45316$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.74 + 9.94i)5-s + (17.2 − 6.73i)7-s + (−49.6 + 28.6i)11-s − 9.20i·13-s + (−14.5 − 25.1i)17-s + (−32.6 − 18.8i)19-s + (7.27 + 4.19i)23-s + (−3.41 − 5.91i)25-s − 62.3i·29-s + (−48.8 + 28.2i)31-s + (−32.1 + 210. i)35-s + (146. − 252. i)37-s − 54.2·41-s − 438.·43-s + (128. − 222. i)47-s + ⋯
L(s)  = 1  + (−0.513 + 0.889i)5-s + (0.931 − 0.363i)7-s + (−1.36 + 0.785i)11-s − 0.196i·13-s + (−0.207 − 0.359i)17-s + (−0.393 − 0.227i)19-s + (0.0659 + 0.0380i)23-s + (−0.0273 − 0.0473i)25-s − 0.399i·29-s + (−0.283 + 0.163i)31-s + (−0.155 + 1.01i)35-s + (0.649 − 1.12i)37-s − 0.206·41-s − 1.55·43-s + (0.398 − 0.690i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.779 + 0.626i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ -0.779 + 0.626i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1989730591\)
\(L(\frac12)\) \(\approx\) \(0.1989730591\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.2 + 6.73i)T \)
good5 \( 1 + (5.74 - 9.94i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (49.6 - 28.6i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 9.20iT - 2.19e3T^{2} \)
17 \( 1 + (14.5 + 25.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (32.6 + 18.8i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-7.27 - 4.19i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 62.3iT - 2.43e4T^{2} \)
31 \( 1 + (48.8 - 28.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-146. + 252. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 54.2T + 6.89e4T^{2} \)
43 \( 1 + 438.T + 7.95e4T^{2} \)
47 \( 1 + (-128. + 222. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (515. - 297. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (238. + 412. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (548. + 316. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (308. + 534. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 396. iT - 3.57e5T^{2} \)
73 \( 1 + (-39.8 + 23.0i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-344. + 597. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.28e3T + 5.71e5T^{2} \)
89 \( 1 + (595. - 1.03e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 946. iT - 9.12e5T^{2} \)
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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

−10.51616064033149772342349860150, −9.334588551210440308186192268777, −8.006273714980522804397824514758, −7.58724626504515315002060042352, −6.68034676039748301862239234267, −5.27359278970785739974360445185, −4.44280116424825653397657304717, −3.12772079536265218055418757866, −1.98721845550348887495819084700, −0.06036498385194223537041098015, 1.43923358075750955003796665677, 2.88046738213082672684760362980, 4.37071388048390428775588839712, 5.09388698264357333430739690792, 6.04719888856771642345747221453, 7.51908309420612203034882907527, 8.378316872633527420144594403267, 8.658146503737057121598054974349, 10.04692266875815745557625417305, 10.98855138528280578833299829523

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