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Properties

Label 2-756-63.31-c2-0-11
Degree $2$
Conductor $756$
Sign $0.224 + 0.974i$
Analytic cond. $20.5995$
Root an. cond. $4.53866$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.39i·5-s + (5.61 − 4.17i)7-s + 2.78·11-s + (18.0 + 10.4i)13-s + (24.1 + 13.9i)17-s + (1.05 − 0.608i)19-s + 16.7·23-s − 45.3·25-s + (−22.8 − 39.6i)29-s + (24.1 − 13.9i)31-s + (−35.0 − 47.1i)35-s + (−19.7 − 34.2i)37-s + (−4.28 − 2.47i)41-s + (31.5 + 54.6i)43-s + (−47.9 − 27.6i)47-s + ⋯
L(s)  = 1  − 1.67i·5-s + (0.802 − 0.596i)7-s + 0.253·11-s + (1.39 + 0.803i)13-s + (1.42 + 0.820i)17-s + (0.0554 − 0.0320i)19-s + 0.727·23-s − 1.81·25-s + (−0.789 − 1.36i)29-s + (0.780 − 0.450i)31-s + (−1.00 − 1.34i)35-s + (−0.534 − 0.925i)37-s + (−0.104 − 0.0603i)41-s + (0.733 + 1.27i)43-s + (−1.01 − 0.588i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.224 + 0.974i$
Analytic conductor: \(20.5995\)
Root analytic conductor: \(4.53866\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1),\ 0.224 + 0.974i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.280379272\)
\(L(\frac12)\) \(\approx\) \(2.280379272\)
\(L(2)\) 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 \( 1 + (-5.61 + 4.17i)T \)
good5 \( 1 + 8.39iT - 25T^{2} \)
11 \( 1 - 2.78T + 121T^{2} \)
13 \( 1 + (-18.0 - 10.4i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-24.1 - 13.9i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.05 + 0.608i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 - 16.7T + 529T^{2} \)
29 \( 1 + (22.8 + 39.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-24.1 + 13.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (19.7 + 34.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (4.28 + 2.47i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-31.5 - 54.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (47.9 + 27.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (8.54 - 14.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (49.2 - 28.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (62.7 + 36.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-30.6 - 53.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 4.84T + 5.04e3T^{2} \)
73 \( 1 + (-64.3 - 37.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-54.4 + 94.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (64.7 - 37.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (72.1 - 41.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (28.1 - 16.2i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.788009035406582856688669189327, −9.025700731134492358575484113694, −8.246812306861965275532219677062, −7.70275983067724130573360488371, −6.25609795508295727685388409059, −5.40468230792560585883547818698, −4.41137265859955579599915629672, −3.77437714937452303497270494821, −1.63573382855981217357162163863, −0.954696881648985550267288735853, 1.41964589221964418758230119922, 2.94791238438941064911326013560, 3.45659873414211148623477010903, 5.10720342044493776997383418600, 5.94177439785079389159484721333, 6.86460338364493785518883312948, 7.69165538032477406307911669618, 8.506227120735162810804700486420, 9.568389481799045634099858895584, 10.58978287883458204493822769300

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