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Properties

Label 2-756-21.17-c3-0-30
Degree $2$
Conductor $756$
Sign $-0.997 + 0.0724i$
Analytic cond. $44.6054$
Root an. cond. $6.67873$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.84 − 15.3i)5-s + (1.00 − 18.4i)7-s + (21.7 + 12.5i)11-s − 88.8i·13-s + (23.1 − 40.0i)17-s + (−37.2 + 21.5i)19-s + (162. − 94.0i)23-s + (−93.9 + 162. i)25-s − 275. i·29-s + (115. + 66.7i)31-s + (−292. + 148. i)35-s + (−50.7 − 87.8i)37-s + 348.·41-s − 266.·43-s + (222. + 384. i)47-s + ⋯
L(s)  = 1  + (−0.791 − 1.37i)5-s + (0.0541 − 0.998i)7-s + (0.596 + 0.344i)11-s − 1.89i·13-s + (0.329 − 0.571i)17-s + (−0.449 + 0.259i)19-s + (1.47 − 0.852i)23-s + (−0.751 + 1.30i)25-s − 1.76i·29-s + (0.669 + 0.386i)31-s + (−1.41 + 0.715i)35-s + (−0.225 − 0.390i)37-s + 1.32·41-s − 0.946·43-s + (0.689 + 1.19i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0724i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.997 + 0.0724i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ -0.997 + 0.0724i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.471100769\)
\(L(\frac12)\) \(\approx\) \(1.471100769\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (-1.00 + 18.4i)T \)
good5 \( 1 + (8.84 + 15.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-21.7 - 12.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 88.8iT - 2.19e3T^{2} \)
17 \( 1 + (-23.1 + 40.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (37.2 - 21.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-162. + 94.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 275. iT - 2.43e4T^{2} \)
31 \( 1 + (-115. - 66.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (50.7 + 87.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 348.T + 6.89e4T^{2} \)
43 \( 1 + 266.T + 7.95e4T^{2} \)
47 \( 1 + (-222. - 384. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (369. + 213. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-21.8 + 37.8i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (286. - 165. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (489. - 847. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 820. iT - 3.57e5T^{2} \)
73 \( 1 + (-404. - 233. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (594. + 1.03e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 251.T + 5.71e5T^{2} \)
89 \( 1 + (-41.8 - 72.4i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 500. iT - 9.12e5T^{2} \)
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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

−9.523186698474629944072521683050, −8.504132759988453776141277662146, −7.911669592671310100000910425445, −7.15520104908458037525004390612, −5.84645192930106990830263601745, −4.75562095882768215890690511810, −4.20910389412214500071055222539, −3.00961388595974616851199502730, −1.06081775482423745845782406066, −0.48695073688285731179257681011, 1.68779761067206672405134672191, 2.93933867506954498065138769509, 3.74851907304801999195858436014, 4.90718122927756614907625153526, 6.30130396552036836649045363577, 6.75954743762819333486297785648, 7.66328669431401097059813544636, 8.837767362798166085362637881098, 9.286827490560484702777419004785, 10.60709985931373188747317828817

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