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Properties

Label 2-756-84.11-c1-0-20
Degree $2$
Conductor $756$
Sign $-0.849 - 0.526i$
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  + (0.373 + 1.36i)2-s + (−1.72 + 1.02i)4-s + (1.28 − 0.743i)5-s + (1.36 + 2.26i)7-s + (−2.03 − 1.96i)8-s + (1.49 + 1.47i)10-s + (−1.37 + 2.37i)11-s − 5.10·13-s + (−2.57 + 2.71i)14-s + (1.91 − 3.50i)16-s + (4.52 + 2.61i)17-s + (−1.30 + 0.754i)19-s + (−1.45 + 2.59i)20-s + (−3.75 − 0.983i)22-s + (4.49 + 7.77i)23-s + ⋯
L(s)  = 1  + (0.264 + 0.964i)2-s + (−0.860 + 0.510i)4-s + (0.575 − 0.332i)5-s + (0.517 + 0.855i)7-s + (−0.719 − 0.694i)8-s + (0.472 + 0.467i)10-s + (−0.413 + 0.716i)11-s − 1.41·13-s + (−0.688 + 0.725i)14-s + (0.479 − 0.877i)16-s + (1.09 + 0.634i)17-s + (−0.299 + 0.172i)19-s + (−0.325 + 0.579i)20-s + (−0.800 − 0.209i)22-s + (0.936 + 1.62i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.849 - 0.526i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.849 - 0.526i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.412496 + 1.44842i\)
\(L(\frac12)\) \(\approx\) \(0.412496 + 1.44842i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.373 - 1.36i)T \)
3 \( 1 \)
7 \( 1 + (-1.36 - 2.26i)T \)
good5 \( 1 + (-1.28 + 0.743i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.37 - 2.37i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.10T + 13T^{2} \)
17 \( 1 + (-4.52 - 2.61i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.30 - 0.754i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.49 - 7.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.11iT - 29T^{2} \)
31 \( 1 + (0.202 + 0.117i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.50 + 4.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.96iT - 41T^{2} \)
43 \( 1 - 6.52iT - 43T^{2} \)
47 \( 1 + (-2.12 - 3.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.44 + 1.98i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.339 - 0.587i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.29 + 9.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.34 + 5.39i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.32T + 71T^{2} \)
73 \( 1 + (1.75 - 3.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-14.4 + 8.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 + (-10.8 + 6.27i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−10.46273394859496779361565883013, −9.434499350517257212613651291245, −9.121525971820686157034512200857, −7.76760135644255931197373198046, −7.48078537225406295205179134079, −6.11504566300760397022298052021, −5.27192367800417951028582761753, −4.87693204607188269115712016779, −3.34375153748230491792107232600, −1.88753550533637379645413645980, 0.71046120064758259769775611524, 2.29872951096354890159363013654, 3.16188619033178959820130215277, 4.53863792692446396736863316292, 5.16829989178780130526862904627, 6.32890593992050810092354963194, 7.47030978227499222900169432299, 8.407588700037103766489697238614, 9.457829866410454047891667328827, 10.36057755245377049198975891243

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