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Properties

Label 2-756-189.5-c1-0-10
Degree $2$
Conductor $756$
Sign $-0.323 - 0.946i$
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  + (1.11 + 1.32i)3-s + (−0.682 + 3.87i)5-s + (2.62 + 0.358i)7-s + (−0.500 + 2.95i)9-s + (0.639 − 0.112i)11-s + (2.79 − 3.32i)13-s + (−5.88 + 3.42i)15-s + 4.23·17-s − 1.27i·19-s + (2.45 + 3.86i)21-s + (−3.69 + 4.40i)23-s + (−9.81 − 3.57i)25-s + (−4.47 + 2.64i)27-s + (−0.340 − 0.405i)29-s + (−1.83 − 5.04i)31-s + ⋯
L(s)  = 1  + (0.645 + 0.763i)3-s + (−0.305 + 1.73i)5-s + (0.990 + 0.135i)7-s + (−0.166 + 0.985i)9-s + (0.192 − 0.0340i)11-s + (0.774 − 0.923i)13-s + (−1.51 + 0.883i)15-s + 1.02·17-s − 0.291i·19-s + (0.535 + 0.844i)21-s + (−0.771 + 0.918i)23-s + (−1.96 − 0.714i)25-s + (−0.860 + 0.508i)27-s + (−0.0632 − 0.0753i)29-s + (−0.329 − 0.905i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.323 - 0.946i$
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: \((2,\ 756,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17529 + 1.64360i\)
\(L(\frac12)\) \(\approx\) \(1.17529 + 1.64360i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 - 1.32i)T \)
7 \( 1 + (-2.62 - 0.358i)T \)
good5 \( 1 + (0.682 - 3.87i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.639 + 0.112i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-2.79 + 3.32i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 - 4.23T + 17T^{2} \)
19 \( 1 + 1.27iT - 19T^{2} \)
23 \( 1 + (3.69 - 4.40i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.340 + 0.405i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.83 + 5.04i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-1.16 - 2.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.14 + 5.15i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.264 + 0.0962i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (8.79 + 3.20i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-8.85 + 5.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.258 - 0.216i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (4.56 - 12.5i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.70 + 9.66i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-9.11 - 5.26i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.00 + 1.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.46 + 13.9i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-12.6 + 10.5i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 4.00T + 89T^{2} \)
97 \( 1 + (-3.15 + 8.67i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.39662353499243527729051428224, −10.10539163295715877171784952868, −8.841461299663608679164696018315, −7.85911893820655628653049686004, −7.51620963928557222913050356305, −6.13974018299871441988895434327, −5.21561537115713312281293385593, −3.80588946483144399342978988096, −3.27336587395752902183856410517, −2.08411173390973540841247139704, 1.08317092980153263932066593195, 1.82522094729200586996136587368, 3.68676044403052120981199154772, 4.54107416226497185302077342945, 5.54112813574110134655911334158, 6.68583494646927669733920836419, 7.903951471765722986454657527706, 8.292706747578907030561470403346, 8.933840400896965775511251842946, 9.782938958466471732426546675293

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