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Properties

Label 2-504-63.38-c1-0-12
Degree $2$
Conductor $504$
Sign $0.967 + 0.252i$
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  + (−1.32 + 1.12i)3-s + (1.38 − 2.39i)5-s + (−1.42 + 2.23i)7-s + (0.486 − 2.96i)9-s + (2.09 − 1.20i)11-s + (−0.0461 + 0.0266i)13-s + (0.860 + 4.71i)15-s + (1.79 − 3.11i)17-s + (3.96 − 2.29i)19-s + (−0.620 − 4.54i)21-s + (0.0925 + 0.0534i)23-s + (−1.33 − 2.31i)25-s + (2.67 + 4.45i)27-s + (6.57 + 3.79i)29-s − 6.81i·31-s + ⋯
L(s)  = 1  + (−0.762 + 0.647i)3-s + (0.619 − 1.07i)5-s + (−0.538 + 0.842i)7-s + (0.162 − 0.986i)9-s + (0.631 − 0.364i)11-s + (−0.0128 + 0.00739i)13-s + (0.222 + 1.21i)15-s + (0.435 − 0.755i)17-s + (0.910 − 0.525i)19-s + (−0.135 − 0.990i)21-s + (0.0193 + 0.0111i)23-s + (−0.266 − 0.462i)25-s + (0.515 + 0.857i)27-s + (1.22 + 0.704i)29-s − 1.22i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20771 - 0.154929i\)
\(L(\frac12)\) \(\approx\) \(1.20771 - 0.154929i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 1.12i)T \)
7 \( 1 + (1.42 - 2.23i)T \)
good5 \( 1 + (-1.38 + 2.39i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.09 + 1.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0461 - 0.0266i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.79 + 3.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.96 + 2.29i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0925 - 0.0534i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.57 - 3.79i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.81iT - 31T^{2} \)
37 \( 1 + (-2.06 - 3.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.838 + 1.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.74 + 3.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 13.1T + 47T^{2} \)
53 \( 1 + (-5.27 - 3.04i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.44T + 59T^{2} \)
61 \( 1 + 2.65iT - 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 - 1.40iT - 71T^{2} \)
73 \( 1 + (-5.54 - 3.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 4.22T + 79T^{2} \)
83 \( 1 + (6.86 - 11.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.71 + 9.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.3 + 8.28i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.89866311503997477886732206733, −9.710209052987802907670573644732, −9.330368809104323357664959943729, −8.599871832666313427336040337504, −7.00670114883253017350273052939, −5.88624229778586867156683143203, −5.38285872201066074756655657694, −4.38193558470382668162861792166, −2.95350378702842308501500918794, −0.976446200599099501788079088579, 1.32382621025594761067462227848, 2.85459680824223467015279139880, 4.19969246396425033391769459182, 5.67400030825091355072409109821, 6.46067772294223124875035437314, 7.06082124198621321751680461196, 7.924739727378360748859917720422, 9.465391438294877140137479336779, 10.42410322548587807675133746564, 10.65600472989571414783629560273

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