Properties

Label 2-756-108.11-c1-0-21
Degree $2$
Conductor $756$
Sign $-0.649 - 0.760i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (−0.270 + 1.38i)2-s + (−1.73 − 0.0414i)3-s + (−1.85 − 0.749i)4-s + (−0.0583 + 0.160i)5-s + (0.525 − 2.39i)6-s + (−0.984 − 0.173i)7-s + (1.54 − 2.37i)8-s + (2.99 + 0.143i)9-s + (−0.206 − 0.124i)10-s + (−1.14 + 0.416i)11-s + (3.17 + 1.37i)12-s + (2.79 − 2.34i)13-s + (0.507 − 1.32i)14-s + (0.107 − 0.275i)15-s + (2.87 + 2.78i)16-s + (−0.976 + 0.563i)17-s + ⋯
L(s)  = 1  + (−0.191 + 0.981i)2-s + (−0.999 − 0.0239i)3-s + (−0.927 − 0.374i)4-s + (−0.0261 + 0.0717i)5-s + (0.214 − 0.976i)6-s + (−0.372 − 0.0656i)7-s + (0.545 − 0.838i)8-s + (0.998 + 0.0477i)9-s + (−0.0654 − 0.0393i)10-s + (−0.344 + 0.125i)11-s + (0.917 + 0.397i)12-s + (0.776 − 0.651i)13-s + (0.135 − 0.352i)14-s + (0.0278 − 0.0710i)15-s + (0.718 + 0.695i)16-s + (−0.236 + 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.649 - 0.760i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.281289 + 0.610651i\)
\(L(\frac12)\) \(\approx\) \(0.281289 + 0.610651i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.270 - 1.38i)T \)
3 \( 1 + (1.73 + 0.0414i)T \)
7 \( 1 + (0.984 + 0.173i)T \)
good5 \( 1 + (0.0583 - 0.160i)T + (-3.83 - 3.21i)T^{2} \)
11 \( 1 + (1.14 - 0.416i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-2.79 + 2.34i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.976 - 0.563i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.62 - 0.936i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.245 - 1.39i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.77 - 3.30i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.93 - 0.516i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.20 + 2.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.85 - 2.21i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.37 - 6.52i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.989 - 5.61i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + (3.41 + 1.24i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.338 + 1.92i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (5.29 + 6.30i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-7.33 - 12.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.29 - 3.98i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.27 - 3.89i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.41 - 1.18i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-12.6 - 7.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.86 + 2.50i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.65527350799973594328439551080, −9.741256755583650663325607088132, −8.951021084606077670471930352724, −7.78758057520136452783246596346, −7.14165173386080172867964786677, −6.17797915663561549268271938230, −5.56774789384325887754779117089, −4.64674954138424784021972409744, −3.47719605459983317096048357565, −1.15588387631105307893873145279, 0.50748401765839711662650296601, 2.00317784128403612567503230839, 3.48854527981671074799669056937, 4.46560569989227405849497876228, 5.38705959598133402223211300671, 6.41559990919981116244164337612, 7.43959069626170667007780208776, 8.624524302210356820014351531812, 9.369724665066910348255820087497, 10.29676193221875596173475904977

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