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Properties

Label 2-756-21.17-c3-0-19
Degree $2$
Conductor $756$
Sign $0.786 - 0.617i$
Analytic cond. $44.6054$
Root an. cond. $6.67873$
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  + (6.73 + 11.6i)5-s + (9.49 + 15.9i)7-s + (−3.36 − 1.94i)11-s − 22.9i·13-s + (57.1 − 99.0i)17-s + (112. − 64.8i)19-s + (145. − 83.8i)23-s + (−28.3 + 49.0i)25-s + 111. i·29-s + (13.4 + 7.76i)31-s + (−121. + 218. i)35-s + (166. + 289. i)37-s + 289.·41-s − 38.3·43-s + (−137. − 238. i)47-s + ⋯
L(s)  = 1  + (0.602 + 1.04i)5-s + (0.512 + 0.858i)7-s + (−0.0921 − 0.0531i)11-s − 0.489i·13-s + (0.815 − 1.41i)17-s + (1.35 − 0.783i)19-s + (1.31 − 0.760i)23-s + (−0.226 + 0.392i)25-s + 0.717i·29-s + (0.0778 + 0.0449i)31-s + (−0.587 + 1.05i)35-s + (0.741 + 1.28i)37-s + 1.10·41-s − 0.136·43-s + (−0.426 − 0.738i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.786 - 0.617i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ 0.786 - 0.617i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.734722184\)
\(L(\frac12)\) \(\approx\) \(2.734722184\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.49 - 15.9i)T \)
good5 \( 1 + (-6.73 - 11.6i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (3.36 + 1.94i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 22.9iT - 2.19e3T^{2} \)
17 \( 1 + (-57.1 + 99.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-112. + 64.8i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-145. + 83.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 111. iT - 2.43e4T^{2} \)
31 \( 1 + (-13.4 - 7.76i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-166. - 289. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 289.T + 6.89e4T^{2} \)
43 \( 1 + 38.3T + 7.95e4T^{2} \)
47 \( 1 + (137. + 238. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (542. + 313. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (232. - 402. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-142. + 81.9i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (11.3 - 19.5i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 740. iT - 3.57e5T^{2} \)
73 \( 1 + (-846. - 488. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-463. - 803. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.15e3T + 5.71e5T^{2} \)
89 \( 1 + (554. + 960. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.07e3iT - 9.12e5T^{2} \)
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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

−9.919707667328177840383027167784, −9.373704804591926980146956348088, −8.324521485671727919631978137250, −7.32686331890499978954361675693, −6.61452154168961561887572183437, −5.46999927281753117610185357327, −4.92703350073728394855115381953, −2.98439370567380480144968305643, −2.71766902783569195964711924581, −1.04974370035105505384419780835, 1.00037843553957202225578148291, 1.67278439189409602414652058132, 3.43984885982983091226534922156, 4.46607503611946287079470250484, 5.35129626406816317406593127864, 6.15364620723520966536320407336, 7.53402100656281103467322563762, 7.993932366564581097717184038949, 9.205956187928528910496890203763, 9.677415253238591983155074883084

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