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Properties

Label 2-114-57.50-c3-0-6
Degree $2$
Conductor $114$
Sign $-0.998 + 0.0495i$
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 + 1.73i)2-s + (0.135 + 5.19i)3-s + (−1.99 + 3.46i)4-s + (−12.0 + 6.93i)5-s + (−8.86 + 5.42i)6-s + 28.4·7-s − 7.99·8-s + (−26.9 + 1.40i)9-s + (−24.0 − 13.8i)10-s + 22.9i·11-s + (−18.2 − 9.91i)12-s + (−70.9 − 40.9i)13-s + (28.4 + 49.3i)14-s + (−37.6 − 61.4i)15-s + (−8 − 13.8i)16-s + (−25.5 + 14.7i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.0260 + 0.999i)3-s + (−0.249 + 0.433i)4-s + (−1.07 + 0.619i)5-s + (−0.602 + 0.369i)6-s + 1.53·7-s − 0.353·8-s + (−0.998 + 0.0520i)9-s + (−0.759 − 0.438i)10-s + 0.627i·11-s + (−0.439 − 0.238i)12-s + (−1.51 − 0.874i)13-s + (0.543 + 0.941i)14-s + (−0.647 − 1.05i)15-s + (−0.125 − 0.216i)16-s + (−0.364 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0495i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.998 + 0.0495i$
Analytic conductor: \(6.72621\)
Root analytic conductor: \(2.59349\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :3/2),\ -0.998 + 0.0495i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0340041 - 1.37221i\)
\(L(\frac12)\) \(\approx\) \(0.0340041 - 1.37221i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 + (-0.135 - 5.19i)T \)
19 \( 1 + (-82.2 + 10.0i)T \)
good5 \( 1 + (12.0 - 6.93i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 - 28.4T + 343T^{2} \)
11 \( 1 - 22.9iT - 1.33e3T^{2} \)
13 \( 1 + (70.9 + 40.9i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (25.5 - 14.7i)T + (2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (-96.1 - 55.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (67.9 - 117. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 278. iT - 2.97e4T^{2} \)
37 \( 1 + 224. iT - 5.06e4T^{2} \)
41 \( 1 + (-216. - 374. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-151. - 261. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-139. - 80.5i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (42.7 - 74.1i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (193. + 335. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (75.6 - 131. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-376. - 217. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (412. + 714. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (115. + 200. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-814. + 470. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 755. iT - 5.71e5T^{2} \)
89 \( 1 + (-654. + 1.13e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (254. - 146. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.25072447614310103665708163571, −12.45372999764854723010196157006, −11.44788730491257753420448339092, −10.69670391952012754696174267437, −9.274606965584146081116930267477, −7.901070886566708273497781206449, −7.34046890886466028713164520214, −5.22221065201717644701030721618, −4.56994198153955183423091486657, −3.09156761968438535610988626294, 0.67456222556268492842457345050, 2.28496109279502990157423358968, 4.29096298518956087809272101005, 5.35721111482930586179108066160, 7.28038355552494178181746380664, 8.088390150911771794043336938269, 9.163788891922485290044569064285, 11.11031749127059912982676155415, 11.74851924602971510909663831164, 12.24570032933135391934106333045

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