Properties

Label 2-504-63.5-c1-0-11
Degree $2$
Conductor $504$
Sign $0.474 + 0.880i$
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  + (−1.68 − 0.397i)3-s + (−0.311 − 0.540i)5-s + (2.62 + 0.362i)7-s + (2.68 + 1.34i)9-s + (−4.50 − 2.59i)11-s + (2.74 + 1.58i)13-s + (0.310 + 1.03i)15-s + (−0.437 − 0.757i)17-s + (−1.41 − 0.819i)19-s + (−4.27 − 1.65i)21-s + (7.14 − 4.12i)23-s + (2.30 − 3.99i)25-s + (−3.99 − 3.32i)27-s + (4.96 − 2.86i)29-s − 4.64i·31-s + ⋯
L(s)  = 1  + (−0.973 − 0.229i)3-s + (−0.139 − 0.241i)5-s + (0.990 + 0.136i)7-s + (0.894 + 0.446i)9-s + (−1.35 − 0.783i)11-s + (0.760 + 0.438i)13-s + (0.0802 + 0.267i)15-s + (−0.106 − 0.183i)17-s + (−0.325 − 0.187i)19-s + (−0.932 − 0.360i)21-s + (1.49 − 0.860i)23-s + (0.461 − 0.798i)25-s + (−0.768 − 0.640i)27-s + (0.921 − 0.532i)29-s − 0.834i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.474 + 0.880i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.474 + 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.891203 - 0.531976i\)
\(L(\frac12)\) \(\approx\) \(0.891203 - 0.531976i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.397i)T \)
7 \( 1 + (-2.62 - 0.362i)T \)
good5 \( 1 + (0.311 + 0.540i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.50 + 2.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.74 - 1.58i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.437 + 0.757i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.41 + 0.819i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.14 + 4.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.96 + 2.86i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.64iT - 31T^{2} \)
37 \( 1 + (-1.24 + 2.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.52 + 6.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.56 + 2.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.46T + 47T^{2} \)
53 \( 1 + (-1.15 + 0.665i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.36T + 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (11.6 - 6.73i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.69T + 79T^{2} \)
83 \( 1 + (-0.192 - 0.332i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.0198 - 0.0344i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.94 - 3.43i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.99270159840321429961453211271, −10.23140012352844488000671336428, −8.735551341617002731038156142541, −8.134733496719862942755690063684, −7.05855743860038512409944707721, −6.00980193912419660252823109476, −5.11443711324922329891842885279, −4.34866712882877988685142175313, −2.47950557171836198805217549288, −0.798416071036907719500102076045, 1.41037335292410485951811077038, 3.24338884464380290807326959146, 4.80067981557185115737216567405, 5.14865822465754880894024530657, 6.45187417972896739178684825605, 7.42108224858010823014266045228, 8.220729503239888679347052807321, 9.474582892899356541494297544814, 10.67060287756099866296519802766, 10.80282855785174891049807304760

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