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Properties

Label 2-504-21.17-c3-0-14
Degree $2$
Conductor $504$
Sign $0.749 + 0.662i$
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  + (3.34 + 5.79i)5-s + (−12.7 + 13.4i)7-s + (−28.2 − 16.3i)11-s − 67.9i·13-s + (15.3 − 26.5i)17-s + (−21.8 + 12.6i)19-s + (68.6 − 39.6i)23-s + (40.0 − 69.4i)25-s − 109. i·29-s + (238. + 137. i)31-s + (−120. − 28.8i)35-s + (160. + 277. i)37-s + 184.·41-s + 364.·43-s + (−25.7 − 44.6i)47-s + ⋯
L(s)  = 1  + (0.299 + 0.518i)5-s + (−0.687 + 0.725i)7-s + (−0.775 − 0.447i)11-s − 1.44i·13-s + (0.218 − 0.378i)17-s + (−0.264 + 0.152i)19-s + (0.622 − 0.359i)23-s + (0.320 − 0.555i)25-s − 0.702i·29-s + (1.38 + 0.797i)31-s + (−0.582 − 0.139i)35-s + (0.711 + 1.23i)37-s + 0.704·41-s + 1.29·43-s + (−0.0799 − 0.138i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.556867096\)
\(L(\frac12)\) \(\approx\) \(1.556867096\)
\(L(\frac{5}{2})\) 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 \( 1 + (12.7 - 13.4i)T \)
good5 \( 1 + (-3.34 - 5.79i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (28.2 + 16.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 67.9iT - 2.19e3T^{2} \)
17 \( 1 + (-15.3 + 26.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (21.8 - 12.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-68.6 + 39.6i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 109. iT - 2.43e4T^{2} \)
31 \( 1 + (-238. - 137. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-160. - 277. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 184.T + 6.89e4T^{2} \)
43 \( 1 - 364.T + 7.95e4T^{2} \)
47 \( 1 + (25.7 + 44.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (532. + 307. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (207. - 359. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-411. + 237. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-142. + 246. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 965. iT - 3.57e5T^{2} \)
73 \( 1 + (-225. - 130. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-219. - 379. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 76.4T + 5.71e5T^{2} \)
89 \( 1 + (356. + 617. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 410. iT - 9.12e5T^{2} \)
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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

−10.33486811720638873616304545051, −9.710888955553363304821715453613, −8.529298512831616775406910483691, −7.81924982112283054586053908035, −6.52695910267521800106900156550, −5.86082737057928078634815774752, −4.84777603557608476827278589978, −3.09817907881282083528802397364, −2.63932637475189635173649075699, −0.56894883648486668263744561171, 1.06724021703635732018806266560, 2.49728071930076682285123941787, 3.94653403162243928045462735057, 4.81866802712378001591965575284, 6.02722310238282653071508722927, 6.97866325765623590269529071315, 7.80471959340018091566477465537, 9.097804121548098939465100260047, 9.574516078240210342767482800638, 10.58930715807409831550479039065

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