Properties

Label 2-114-19.5-c3-0-1
Degree $2$
Conductor $114$
Sign $0.914 - 0.405i$
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  + (−1.87 + 0.684i)2-s + (0.520 − 2.95i)3-s + (3.06 − 2.57i)4-s + (16.3 + 13.7i)5-s + (1.04 + 5.90i)6-s + (−6.08 − 10.5i)7-s + (−4.00 + 6.92i)8-s + (−8.45 − 3.07i)9-s + (−40.2 − 14.6i)10-s + (1.05 − 1.83i)11-s + (−6.00 − 10.3i)12-s + (10.9 + 62.3i)13-s + (18.6 + 15.6i)14-s + (49.1 − 41.2i)15-s + (2.77 − 15.7i)16-s + (83.9 − 30.5i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.100 − 0.568i)3-s + (0.383 − 0.321i)4-s + (1.46 + 1.22i)5-s + (0.0708 + 0.402i)6-s + (−0.328 − 0.569i)7-s + (−0.176 + 0.306i)8-s + (−0.313 − 0.114i)9-s + (−1.27 − 0.462i)10-s + (0.0290 − 0.0502i)11-s + (−0.144 − 0.250i)12-s + (0.234 + 1.32i)13-s + (0.355 + 0.298i)14-s + (0.846 − 0.710i)15-s + (0.0434 − 0.246i)16-s + (1.19 − 0.435i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.914 - 0.405i$
Analytic conductor: \(6.72621\)
Root analytic conductor: \(2.59349\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :3/2),\ 0.914 - 0.405i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.48831 + 0.315241i\)
\(L(\frac12)\) \(\approx\) \(1.48831 + 0.315241i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.87 - 0.684i)T \)
3 \( 1 + (-0.520 + 2.95i)T \)
19 \( 1 + (-81.9 - 11.9i)T \)
good5 \( 1 + (-16.3 - 13.7i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (6.08 + 10.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-1.05 + 1.83i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-10.9 - 62.3i)T + (-2.06e3 + 751. i)T^{2} \)
17 \( 1 + (-83.9 + 30.5i)T + (3.76e3 - 3.15e3i)T^{2} \)
23 \( 1 + (-57.4 + 48.2i)T + (2.11e3 - 1.19e4i)T^{2} \)
29 \( 1 + (-36.6 - 13.3i)T + (1.86e4 + 1.56e4i)T^{2} \)
31 \( 1 + (7.81 + 13.5i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 98.1T + 5.06e4T^{2} \)
41 \( 1 + (-6.55 + 37.1i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (296. + 248. i)T + (1.38e4 + 7.82e4i)T^{2} \)
47 \( 1 + (-276. - 100. i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 + (-97.0 + 81.4i)T + (2.58e4 - 1.46e5i)T^{2} \)
59 \( 1 + (538. - 195. i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (562. - 472. i)T + (3.94e4 - 2.23e5i)T^{2} \)
67 \( 1 + (779. + 283. i)T + (2.30e5 + 1.93e5i)T^{2} \)
71 \( 1 + (-340. - 285. i)T + (6.21e4 + 3.52e5i)T^{2} \)
73 \( 1 + (-197. + 1.11e3i)T + (-3.65e5 - 1.33e5i)T^{2} \)
79 \( 1 + (215. - 1.22e3i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (363. + 628. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (20.3 + 115. i)T + (-6.62e5 + 2.41e5i)T^{2} \)
97 \( 1 + (1.01e3 - 370. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−13.75043871606385627687887059890, −12.00635859790825282007240746571, −10.78768650594064716250208993088, −9.932337222723878657572404359494, −9.092380816083932674509960926464, −7.33796709426623359776535684290, −6.73483395070304794291444099738, −5.68221426350017298151014767792, −3.02820642080051744315865180988, −1.54462573773181508691420225049, 1.22822249945304653490391551145, 3.00578414075128094742081324263, 5.18481819827936241079208228221, 5.94668965146271388829850914684, 8.009858239972815074210020278636, 9.049713552704330188023477622754, 9.722610176686083315515251368082, 10.49443627017403603569174467170, 12.11959364941839936513941164196, 12.90353590086803024578464517046

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