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Properties

Label 2-504-21.17-c3-0-16
Degree $2$
Conductor $504$
Sign $0.936 + 0.351i$
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  + (7.29 + 12.6i)5-s + (18.4 + 0.998i)7-s + (−38.7 − 22.3i)11-s − 76.8i·13-s + (58.3 − 101. i)17-s + (83.5 − 48.2i)19-s + (6.88 − 3.97i)23-s + (−43.9 + 76.1i)25-s + 86.8i·29-s + (216. + 124. i)31-s + (122. + 240. i)35-s + (−160. − 277. i)37-s − 231.·41-s − 413.·43-s + (235. + 407. i)47-s + ⋯
L(s)  = 1  + (0.652 + 1.13i)5-s + (0.998 + 0.0539i)7-s + (−1.06 − 0.612i)11-s − 1.63i·13-s + (0.832 − 1.44i)17-s + (1.00 − 0.582i)19-s + (0.0624 − 0.0360i)23-s + (−0.351 + 0.608i)25-s + 0.556i·29-s + (1.25 + 0.723i)31-s + (0.590 + 1.16i)35-s + (−0.711 − 1.23i)37-s − 0.881·41-s − 1.46·43-s + (0.730 + 1.26i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.936 + 0.351i$
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: \((2,\ 504,\ (\ :3/2),\ 0.936 + 0.351i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.398617013\)
\(L(\frac12)\) \(\approx\) \(2.398617013\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.4 - 0.998i)T \)
good5 \( 1 + (-7.29 - 12.6i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (38.7 + 22.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 76.8iT - 2.19e3T^{2} \)
17 \( 1 + (-58.3 + 101. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-83.5 + 48.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-6.88 + 3.97i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 86.8iT - 2.43e4T^{2} \)
31 \( 1 + (-216. - 124. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (160. + 277. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 231.T + 6.89e4T^{2} \)
43 \( 1 + 413.T + 7.95e4T^{2} \)
47 \( 1 + (-235. - 407. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-600. - 346. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-143. + 249. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-740. + 427. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-240. + 416. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 930. iT - 3.57e5T^{2} \)
73 \( 1 + (98.4 + 56.8i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-111. - 192. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 692.T + 5.71e5T^{2} \)
89 \( 1 + (258. + 448. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 807. iT - 9.12e5T^{2} \)
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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

−10.46467670464860046317388022306, −9.877378326992060451467101604966, −8.522690754326897694498663436938, −7.68799425789336564788899308448, −6.94155216736175513985202827842, −5.43924505030054673479640733661, −5.24046309611847892857173555723, −3.15843813147838198262909613693, −2.62156624845615416061105684095, −0.833106655047666592491770748006, 1.29736593259197948195922589484, 2.06861603833603365925559401708, 4.00373739018029341987412031321, 4.98661536134711357198866498009, 5.60492737624533719816726985487, 6.94473300784686390685339650258, 8.144012128881926891022211901968, 8.578400853405396113942784645086, 9.842536898019145812071984168181, 10.25300984248054881898077168993

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