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Properties

Label 2-756-84.23-c1-0-27
Degree $2$
Conductor $756$
Sign $0.916 + 0.400i$
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  + (−0.943 − 1.05i)2-s + (−0.219 + 1.98i)4-s + (2.75 + 1.59i)5-s + (0.944 − 2.47i)7-s + (2.30 − 1.64i)8-s + (−0.924 − 4.40i)10-s + (1.24 + 2.16i)11-s + 6.61·13-s + (−3.49 + 1.33i)14-s + (−3.90 − 0.874i)16-s + (−2.67 + 1.54i)17-s + (−4.40 − 2.54i)19-s + (−3.77 + 5.13i)20-s + (1.10 − 3.35i)22-s + (−2.53 + 4.39i)23-s + ⋯
L(s)  = 1  + (−0.667 − 0.744i)2-s + (−0.109 + 0.993i)4-s + (1.23 + 0.712i)5-s + (0.356 − 0.934i)7-s + (0.813 − 0.581i)8-s + (−0.292 − 1.39i)10-s + (0.376 + 0.651i)11-s + 1.83·13-s + (−0.933 + 0.357i)14-s + (−0.975 − 0.218i)16-s + (−0.649 + 0.374i)17-s + (−1.01 − 0.583i)19-s + (−0.843 + 1.14i)20-s + (0.234 − 0.715i)22-s + (−0.529 + 0.916i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.916 + 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45520 - 0.303860i\)
\(L(\frac12)\) \(\approx\) \(1.45520 - 0.303860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.943 + 1.05i)T \)
3 \( 1 \)
7 \( 1 + (-0.944 + 2.47i)T \)
good5 \( 1 + (-2.75 - 1.59i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.24 - 2.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.61T + 13T^{2} \)
17 \( 1 + (2.67 - 1.54i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.40 + 2.54i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.53 - 4.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.59iT - 29T^{2} \)
31 \( 1 + (-7.31 + 4.22i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.357 - 0.619i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 12.2iT - 41T^{2} \)
43 \( 1 - 3.72iT - 43T^{2} \)
47 \( 1 + (-2.51 + 4.35i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.21 + 4.16i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.36 + 9.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.997 + 1.72i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.00277 + 0.00160i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + (-0.957 - 1.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.11 + 3.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 + (-0.482 - 0.278i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.127T + 97T^{2} \)
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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

−10.17256051367361836088794450289, −9.765407390871698021508163551482, −8.706586779062018880469049310539, −7.919416306106086978002142298075, −6.70323872766911125675095087060, −6.24733994365550898117540218375, −4.51142940467637518378279012864, −3.63647841138880381499730144092, −2.27247187422275117036617849884, −1.34045519686839798341427565493, 1.20652809554555796528684789310, 2.26091114785478285435511087979, 4.26982078859642000398223128709, 5.47882441797517185249633019019, 6.00422053352836114928802951006, 6.62632887687353795886412449649, 8.272047262234213416294352144411, 8.781193979568525224643252524878, 9.067631046120878628894744017118, 10.32396877668352956917472027908

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