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Properties

Label 2-504-56.19-c1-0-32
Degree $2$
Conductor $504$
Sign $0.948 + 0.316i$
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  + (1.40 + 0.141i)2-s + (1.95 + 0.398i)4-s + (1.91 − 3.32i)5-s + (1.55 + 2.13i)7-s + (2.70 + 0.838i)8-s + (3.17 − 4.40i)10-s + (−1.28 − 2.21i)11-s − 5.99·13-s + (1.88 + 3.23i)14-s + (3.68 + 1.56i)16-s + (−3.53 + 2.04i)17-s + (−2.05 − 1.18i)19-s + (5.08 − 5.75i)20-s + (−1.48 − 3.30i)22-s + (6.53 + 3.77i)23-s + ⋯
L(s)  = 1  + (0.994 + 0.100i)2-s + (0.979 + 0.199i)4-s + (0.858 − 1.48i)5-s + (0.588 + 0.808i)7-s + (0.955 + 0.296i)8-s + (1.00 − 1.39i)10-s + (−0.386 − 0.669i)11-s − 1.66·13-s + (0.504 + 0.863i)14-s + (0.920 + 0.390i)16-s + (−0.857 + 0.494i)17-s + (−0.471 − 0.271i)19-s + (1.13 − 1.28i)20-s + (−0.317 − 0.704i)22-s + (1.36 + 0.787i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.948 + 0.316i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.91636 - 0.473477i\)
\(L(\frac12)\) \(\approx\) \(2.91636 - 0.473477i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.141i)T \)
3 \( 1 \)
7 \( 1 + (-1.55 - 2.13i)T \)
good5 \( 1 + (-1.91 + 3.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.28 + 2.21i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.99T + 13T^{2} \)
17 \( 1 + (3.53 - 2.04i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.05 + 1.18i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.53 - 3.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.70iT - 29T^{2} \)
31 \( 1 + (-2.90 - 5.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.61 - 2.08i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.96iT - 41T^{2} \)
43 \( 1 + 8.06T + 43T^{2} \)
47 \( 1 + (0.204 - 0.353i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.41 - 2.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.05 + 5.22i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.34 - 2.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.38 - 4.12i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.21iT - 71T^{2} \)
73 \( 1 + (6.65 - 3.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.89 + 5.13i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.49iT - 83T^{2} \)
89 \( 1 + (3.35 + 1.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.20iT - 97T^{2} \)
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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

−11.14540545367406794900061835794, −10.01072692961315061926455132769, −8.919274317310417817263053762730, −8.316419643568197783018275480679, −7.03655507295791761983405911988, −5.81921573795677560274888896731, −5.08650966421566549419130358128, −4.64271395307232573393812174296, −2.76740401303240499659542005943, −1.68999005101165723922148787428, 2.16065142264301150804498126744, 2.80372644576505468808331846242, 4.36637895241554006837660965671, 5.14243600509561147498504140179, 6.50745724145361036881763566802, 7.01462587512193275875788976234, 7.77556861888318354172372434495, 9.725406654998029799984589581770, 10.24589643083508787627874613630, 11.00672910838846048945803639466

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