Properties

Label 2-114-19.16-c3-0-6
Degree $2$
Conductor $114$
Sign $-0.146 + 0.989i$
Analytic cond. $6.72621$
Root an. cond. $2.59349$
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  + (−1.53 − 1.28i)2-s + (2.81 − 1.02i)3-s + (0.694 + 3.93i)4-s + (−0.778 + 4.41i)5-s + (−5.63 − 2.05i)6-s + (−10.3 − 18.0i)7-s + (4.00 − 6.92i)8-s + (6.89 − 5.78i)9-s + (6.86 − 5.76i)10-s + (26.1 − 45.2i)11-s + (6 + 10.3i)12-s + (44.9 + 16.3i)13-s + (−7.22 + 40.9i)14-s + (2.33 + 13.2i)15-s + (−15.0 + 5.47i)16-s + (−103. − 86.6i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.542 − 0.197i)3-s + (0.0868 + 0.492i)4-s + (−0.0696 + 0.394i)5-s + (−0.383 − 0.139i)6-s + (−0.561 − 0.972i)7-s + (0.176 − 0.306i)8-s + (0.255 − 0.214i)9-s + (0.217 − 0.182i)10-s + (0.716 − 1.24i)11-s + (0.144 + 0.249i)12-s + (0.957 + 0.348i)13-s + (−0.137 + 0.781i)14-s + (0.0401 + 0.227i)15-s + (−0.234 + 0.0855i)16-s + (−1.47 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.146 + 0.989i$
Analytic conductor: \(6.72621\)
Root analytic conductor: \(2.59349\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :3/2),\ -0.146 + 0.989i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.845081 - 0.979118i\)
\(L(\frac12)\) \(\approx\) \(0.845081 - 0.979118i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.53 + 1.28i)T \)
3 \( 1 + (-2.81 + 1.02i)T \)
19 \( 1 + (-77.4 + 29.3i)T \)
good5 \( 1 + (0.778 - 4.41i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (10.3 + 18.0i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-26.1 + 45.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-44.9 - 16.3i)T + (1.68e3 + 1.41e3i)T^{2} \)
17 \( 1 + (103. + 86.6i)T + (853. + 4.83e3i)T^{2} \)
23 \( 1 + (25.3 + 143. i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (197. - 165. i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (82.3 + 142. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 157.T + 5.06e4T^{2} \)
41 \( 1 + (-293. + 106. i)T + (5.27e4 - 4.43e4i)T^{2} \)
43 \( 1 + (85.9 - 487. i)T + (-7.47e4 - 2.71e4i)T^{2} \)
47 \( 1 + (167. - 140. i)T + (1.80e4 - 1.02e5i)T^{2} \)
53 \( 1 + (-12.6 - 71.6i)T + (-1.39e5 + 5.09e4i)T^{2} \)
59 \( 1 + (-580. - 486. i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (-89.5 - 507. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (-211. + 177. i)T + (5.22e4 - 2.96e5i)T^{2} \)
71 \( 1 + (-80.7 + 457. i)T + (-3.36e5 - 1.22e5i)T^{2} \)
73 \( 1 + (222. - 80.8i)T + (2.98e5 - 2.50e5i)T^{2} \)
79 \( 1 + (-476. + 173. i)T + (3.77e5 - 3.16e5i)T^{2} \)
83 \( 1 + (-76.1 - 131. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (447. + 162. i)T + (5.40e5 + 4.53e5i)T^{2} \)
97 \( 1 + (1.42e3 + 1.19e3i)T + (1.58e5 + 8.98e5i)T^{2} \)
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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

−13.05971327108497455031363700496, −11.35695583603189907986413246118, −10.92776563810883347586373630809, −9.434082019695207042459727778130, −8.762107494474369424913531139081, −7.32331188457808345880017480125, −6.46837259977885058064359528522, −4.03917118951440531749051676019, −2.91327719590125125527810372878, −0.827281128377350549530937190270, 1.87634026131503754535326379396, 3.90178003536337013879083866606, 5.58599591482887062901067042033, 6.81476313011499266796197739618, 8.193395720397781730740169252475, 9.102757327204941391499719529504, 9.760749565422531795184168943896, 11.19614607187413256758109199292, 12.48934197773672478293839703726, 13.36889264765434804016210849951

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