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Properties

Label 2-756-252.115-c1-0-33
Degree $2$
Conductor $756$
Sign $-0.959 + 0.281i$
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  + (−1.06 − 0.928i)2-s + (0.275 + 1.98i)4-s + (−2.33 − 1.34i)5-s + (1.11 − 2.39i)7-s + (1.54 − 2.36i)8-s + (1.23 + 3.60i)10-s + (1.22 − 0.706i)11-s + (2.46 − 1.42i)13-s + (−3.41 + 1.51i)14-s + (−3.84 + 1.09i)16-s + (1.23 + 0.712i)17-s + (1.48 + 2.56i)19-s + (2.02 − 4.98i)20-s + (−1.96 − 0.382i)22-s + (−1.67 − 0.967i)23-s + ⋯
L(s)  = 1  + (−0.754 − 0.656i)2-s + (0.137 + 0.990i)4-s + (−1.04 − 0.602i)5-s + (0.422 − 0.906i)7-s + (0.546 − 0.837i)8-s + (0.391 + 1.13i)10-s + (0.369 − 0.213i)11-s + (0.683 − 0.394i)13-s + (−0.914 + 0.405i)14-s + (−0.962 + 0.272i)16-s + (0.299 + 0.172i)17-s + (0.339 + 0.588i)19-s + (0.452 − 1.11i)20-s + (−0.418 − 0.0816i)22-s + (−0.349 − 0.201i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.959 + 0.281i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.959 + 0.281i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0919275 - 0.639070i\)
\(L(\frac12)\) \(\approx\) \(0.0919275 - 0.639070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.06 + 0.928i)T \)
3 \( 1 \)
7 \( 1 + (-1.11 + 2.39i)T \)
good5 \( 1 + (2.33 + 1.34i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.22 + 0.706i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.46 + 1.42i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.23 - 0.712i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.48 - 2.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.67 + 0.967i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.423 - 0.733i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.87T + 31T^{2} \)
37 \( 1 + (5.49 + 9.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.99 + 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.96 + 3.44i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + (-6.58 + 11.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.42T + 59T^{2} \)
61 \( 1 - 2.03iT - 61T^{2} \)
67 \( 1 + 8.74iT - 67T^{2} \)
71 \( 1 + 1.95iT - 71T^{2} \)
73 \( 1 + (-7.16 - 4.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 + (2.85 - 4.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.75 - 3.32i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.1 - 7.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.05662569134563973700529187567, −9.020826649490178359864271041156, −8.240165026277886259089599911231, −7.71731513785781943339650274728, −6.83988401182165854970191974120, −5.28483797502654407828980425191, −3.82201340415953043142082942878, −3.72479411337191078011073383606, −1.68525005949753302473465541141, −0.44590882654942545002448493670, 1.64319615327416564046415384487, 3.18605787172803650169670612345, 4.53258133885561283008472822501, 5.59922906127548624779200108722, 6.57110391679272262761590531810, 7.38013544820999793063762061997, 8.125763469979066774342972463826, 8.902179184478142634028081159749, 9.647242273505744613299627705158, 10.76889661018918736878969540128

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