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Properties

Label 2-756-189.5-c1-0-4
Degree $2$
Conductor $756$
Sign $-0.943 + 0.330i$
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  + (−0.538 + 1.64i)3-s + (−0.430 + 2.44i)5-s + (−0.571 + 2.58i)7-s + (−2.42 − 1.77i)9-s + (−0.859 + 0.151i)11-s + (−0.741 + 0.883i)13-s + (−3.79 − 2.02i)15-s − 1.65·17-s + 1.94i·19-s + (−3.94 − 2.33i)21-s + (3.26 − 3.88i)23-s + (−1.08 − 0.395i)25-s + (4.21 − 3.03i)27-s + (−2.90 − 3.46i)29-s + (1.64 + 4.52i)31-s + ⋯
L(s)  = 1  + (−0.310 + 0.950i)3-s + (−0.192 + 1.09i)5-s + (−0.216 + 0.976i)7-s + (−0.807 − 0.590i)9-s + (−0.259 + 0.0456i)11-s + (−0.205 + 0.245i)13-s + (−0.978 − 0.522i)15-s − 0.400·17-s + 0.445i·19-s + (−0.860 − 0.508i)21-s + (0.679 − 0.810i)23-s + (−0.217 − 0.0790i)25-s + (0.812 − 0.583i)27-s + (−0.540 − 0.643i)29-s + (0.295 + 0.812i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.943 + 0.330i$
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: \((2,\ 756,\ (\ :1/2),\ -0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124871 - 0.734644i\)
\(L(\frac12)\) \(\approx\) \(0.124871 - 0.734644i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.538 - 1.64i)T \)
7 \( 1 + (0.571 - 2.58i)T \)
good5 \( 1 + (0.430 - 2.44i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (0.859 - 0.151i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (0.741 - 0.883i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + 1.65T + 17T^{2} \)
19 \( 1 - 1.94iT - 19T^{2} \)
23 \( 1 + (-3.26 + 3.88i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (2.90 + 3.46i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.64 - 4.52i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (2.87 + 4.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.42 + 1.19i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-4.32 - 1.57i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (6.51 + 2.36i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (5.19 - 2.99i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.36 - 7.02i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.06 - 5.66i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.01 - 11.4i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (14.0 + 8.13i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.46 - 0.845i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.801 - 4.54i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.417 + 0.350i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 9.52T + 89T^{2} \)
97 \( 1 + (-1.27 + 3.50i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.71926493886683226415780805829, −10.12276325523270446784279199124, −9.175109827057606275360204577202, −8.493730295345925412601765325649, −7.20354317375962149591647617382, −6.32811739617944537414406275537, −5.50978559063919787775245465455, −4.45999729912967967102738417657, −3.30021549798553103097126595370, −2.47707682887432181733886969858, 0.39293032463527712493210098440, 1.56566166500276430735316300997, 3.17289450562364021902110361766, 4.58144468654072596896305598335, 5.31227092730914491543376525736, 6.47937001790718787390982203629, 7.31127521240683431056649319554, 8.029661981958013206363218100891, 8.871733430950789777833181055934, 9.836038456472524881272464618766

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