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Properties

Label 2-234-117.11-c1-0-4
Degree $2$
Conductor $234$
Sign $0.112 + 0.993i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
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  + (−0.707 + 0.707i)2-s + (−1.57 + 0.726i)3-s − 1.00i·4-s + (0.653 + 0.175i)5-s + (0.597 − 1.62i)6-s + (−3.90 − 1.04i)7-s + (0.707 + 0.707i)8-s + (1.94 − 2.28i)9-s + (−0.585 + 0.338i)10-s + (−0.502 − 0.502i)11-s + (0.726 + 1.57i)12-s + (−2.29 − 2.77i)13-s + (3.49 − 2.01i)14-s + (−1.15 + 0.199i)15-s − 1.00·16-s + (3.24 − 5.61i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.907 + 0.419i)3-s − 0.500i·4-s + (0.292 + 0.0782i)5-s + (0.243 − 0.663i)6-s + (−1.47 − 0.394i)7-s + (0.250 + 0.250i)8-s + (0.647 − 0.761i)9-s + (−0.185 + 0.106i)10-s + (−0.151 − 0.151i)11-s + (0.209 + 0.453i)12-s + (−0.637 − 0.770i)13-s + (0.934 − 0.539i)14-s + (−0.297 + 0.0515i)15-s − 0.250·16-s + (0.786 − 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.112 + 0.993i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.112 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.250040 - 0.223345i\)
\(L(\frac12)\) \(\approx\) \(0.250040 - 0.223345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (1.57 - 0.726i)T \)
13 \( 1 + (2.29 + 2.77i)T \)
good5 \( 1 + (-0.653 - 0.175i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (3.90 + 1.04i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.502 + 0.502i)T + 11iT^{2} \)
17 \( 1 + (-3.24 + 5.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.253 - 0.0679i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.860 + 1.49i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.28iT - 29T^{2} \)
31 \( 1 + (-1.04 + 3.89i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (7.96 + 2.13i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.39 - 8.94i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.67 - 3.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.07 - 2.43i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 6.34iT - 53T^{2} \)
59 \( 1 + (3.52 + 3.52i)T + 59iT^{2} \)
61 \( 1 + (1.64 + 2.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.0 + 2.70i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.09 + 7.82i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.40 - 1.40i)T - 73iT^{2} \)
79 \( 1 + (-7.29 + 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.04 - 3.90i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (2.64 - 9.85i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.520 + 1.94i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.89525451707774612525624335150, −10.69794958110887916538089510891, −9.755552471637068486689899489360, −9.591164822935936238823358703334, −7.79249875025329875504804660060, −6.74668856679165901609556055885, −5.96303974874878413972181553434, −4.88162305254278628553277744137, −3.19257265030370646425806116992, −0.34570584695317748369170126342, 1.88096976251078727627822718868, 3.58901930326341224334228308131, 5.32195317498869915273335430380, 6.41056625244526405319067680414, 7.25333292911417480518355737318, 8.658543761590445363589947704119, 9.850453201472966548107026419473, 10.26816080993993132451837487572, 11.55310210325390548217440494883, 12.40925791854938451744987758951

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