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Properties

Label 2-756-189.5-c1-0-13
Degree $2$
Conductor $756$
Sign $0.986 - 0.164i$
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  + (1.37 − 1.05i)3-s + (−0.403 + 2.29i)5-s + (2.60 + 0.435i)7-s + (0.762 − 2.90i)9-s + (1.79 − 0.315i)11-s + (−2.58 + 3.07i)13-s + (1.86 + 3.56i)15-s + 1.25·17-s + 5.76i·19-s + (4.04 − 2.16i)21-s + (5.50 − 6.55i)23-s + (−0.386 − 0.140i)25-s + (−2.02 − 4.78i)27-s + (−4.11 − 4.90i)29-s + (2.63 + 7.24i)31-s + ⋯
L(s)  = 1  + (0.791 − 0.610i)3-s + (−0.180 + 1.02i)5-s + (0.986 + 0.164i)7-s + (0.254 − 0.967i)9-s + (0.540 − 0.0952i)11-s + (−0.715 + 0.853i)13-s + (0.482 + 0.921i)15-s + 0.304·17-s + 1.32i·19-s + (0.881 − 0.471i)21-s + (1.14 − 1.36i)23-s + (−0.0772 − 0.0281i)25-s + (−0.389 − 0.921i)27-s + (−0.763 − 0.909i)29-s + (0.473 + 1.30i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.986 - 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17321 + 0.180489i\)
\(L(\frac12)\) \(\approx\) \(2.17321 + 0.180489i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.37 + 1.05i)T \)
7 \( 1 + (-2.60 - 0.435i)T \)
good5 \( 1 + (0.403 - 2.29i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-1.79 + 0.315i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (2.58 - 3.07i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 - 1.25T + 17T^{2} \)
19 \( 1 - 5.76iT - 19T^{2} \)
23 \( 1 + (-5.50 + 6.55i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (4.11 + 4.90i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-2.63 - 7.24i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-2.04 - 3.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.98 + 1.66i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (4.77 + 1.73i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-5.64 - 2.05i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.91 - 1.68i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.51 + 6.30i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-3.74 + 10.2i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.64 + 9.31i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-4.63 - 2.67i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.35 + 4.24i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.709 - 4.02i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (7.74 - 6.49i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 6.53T + 89T^{2} \)
97 \( 1 + (-3.30 + 9.08i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.37355632235761949699538946348, −9.404365749849942333457164610475, −8.515840361488759648624627218034, −7.77942015151458666556826830740, −6.95731842944756037671432114218, −6.31110820007558771744448059920, −4.82044924091812094629450644266, −3.68771356635128811012034244328, −2.63096476183216877262751516865, −1.56508703770219932223642119649, 1.24647308894434605008719075368, 2.71239996518504083092787017511, 3.98067062256672830627434999207, 4.88649577375936077218835747220, 5.38126094102885512392803916926, 7.26984668209240404288826285337, 7.80092980419356000170179118441, 8.827976907505879928275983320339, 9.201683310366092702973897825399, 10.19385299944550930831539089404

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