Properties

Label 2-504-21.17-c3-0-12
Degree $2$
Conductor $504$
Sign $0.997 - 0.0755i$
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  + (−1.56 − 2.70i)5-s + (17.1 − 6.88i)7-s + (33.2 + 19.2i)11-s + 13.8i·13-s + (−47.5 + 82.4i)17-s + (9.86 − 5.69i)19-s + (23.9 − 13.8i)23-s + (57.6 − 99.7i)25-s − 44.2i·29-s + (−119. − 68.7i)31-s + (−45.4 − 35.7i)35-s + (70.6 + 122. i)37-s + 337.·41-s + 417.·43-s + (145. + 251. i)47-s + ⋯
L(s)  = 1  + (−0.139 − 0.242i)5-s + (0.928 − 0.371i)7-s + (0.912 + 0.526i)11-s + 0.294i·13-s + (−0.678 + 1.17i)17-s + (0.119 − 0.0687i)19-s + (0.217 − 0.125i)23-s + (0.460 − 0.798i)25-s − 0.283i·29-s + (−0.690 − 0.398i)31-s + (−0.219 − 0.172i)35-s + (0.313 + 0.543i)37-s + 1.28·41-s + 1.48·43-s + (0.450 + 0.779i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0755i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.997 - 0.0755i$
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: \((2,\ 504,\ (\ :3/2),\ 0.997 - 0.0755i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.237486389\)
\(L(\frac12)\) \(\approx\) \(2.237486389\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.1 + 6.88i)T \)
good5 \( 1 + (1.56 + 2.70i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-33.2 - 19.2i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 13.8iT - 2.19e3T^{2} \)
17 \( 1 + (47.5 - 82.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-9.86 + 5.69i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-23.9 + 13.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 44.2iT - 2.43e4T^{2} \)
31 \( 1 + (119. + 68.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-70.6 - 122. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 337.T + 6.89e4T^{2} \)
43 \( 1 - 417.T + 7.95e4T^{2} \)
47 \( 1 + (-145. - 251. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (14.7 + 8.53i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (299. - 519. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-459. + 265. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-325. + 563. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 934. iT - 3.57e5T^{2} \)
73 \( 1 + (-787. - 454. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (397. + 688. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 314.T + 5.71e5T^{2} \)
89 \( 1 + (-179. - 310. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 80.5iT - 9.12e5T^{2} \)
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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

−10.70473124900923016384336846351, −9.541812801372215699837996112096, −8.717216350870893189872805265802, −7.86116489569989822129386138831, −6.90631028391519219496157662035, −5.90373479921296156657630167212, −4.53425881519571596129073551201, −4.03870793861668848659657037504, −2.21632915276568014050766557021, −1.03898276460513388204614974726, 0.926702944733652762310607492056, 2.37277446195190318159171857325, 3.65413981621176127171894148691, 4.84201424516975010696387138859, 5.75961615076726083027118917677, 6.94460331607877295118164012830, 7.73232533208892094126600104176, 8.883000919946239433311830276639, 9.337649797073972603102518330856, 10.83635085634328437088345781640

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