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Properties

Label 2-234-117.32-c1-0-6
Degree $2$
Conductor $234$
Sign $0.182 - 0.983i$
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 + (1.25 + 1.19i)3-s + 1.00i·4-s + (0.650 − 0.174i)5-s + (0.0444 + 1.73i)6-s + (−1.73 + 0.463i)7-s + (−0.707 + 0.707i)8-s + (0.154 + 2.99i)9-s + (0.583 + 0.336i)10-s + (0.696 − 0.696i)11-s + (−1.19 + 1.25i)12-s + (0.391 − 3.58i)13-s + (−1.55 − 0.895i)14-s + (1.02 + 0.557i)15-s − 1.00·16-s + (−1.20 − 2.09i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.725 + 0.688i)3-s + 0.500i·4-s + (0.290 − 0.0779i)5-s + (0.0181 + 0.706i)6-s + (−0.654 + 0.175i)7-s + (−0.250 + 0.250i)8-s + (0.0513 + 0.998i)9-s + (0.184 + 0.106i)10-s + (0.210 − 0.210i)11-s + (−0.344 + 0.362i)12-s + (0.108 − 0.994i)13-s + (−0.414 − 0.239i)14-s + (0.264 + 0.143i)15-s − 0.250·16-s + (−0.293 − 0.508i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.182 - 0.983i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.182 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46213 + 1.21586i\)
\(L(\frac12)\) \(\approx\) \(1.46213 + 1.21586i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 - 1.19i)T \)
13 \( 1 + (-0.391 + 3.58i)T \)
good5 \( 1 + (-0.650 + 0.174i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.73 - 0.463i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.696 + 0.696i)T - 11iT^{2} \)
17 \( 1 + (1.20 + 2.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.33 - 1.69i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.16 + 5.47i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.29iT - 29T^{2} \)
31 \( 1 + (-1.49 - 5.59i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.83 + 1.83i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.28 + 4.80i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.772 + 0.446i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.66 + 0.981i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 9.00iT - 53T^{2} \)
59 \( 1 + (3.33 - 3.33i)T - 59iT^{2} \)
61 \( 1 + (2.38 - 4.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.6 - 2.86i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.70 - 6.37i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (8.01 + 8.01i)T + 73iT^{2} \)
79 \( 1 + (0.807 + 1.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.95 - 7.31i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.440 + 1.64i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.497 + 1.85i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.66529339761032358027485095795, −11.50476291713381441474012759334, −10.19131057951661639564293872030, −9.488977467792132742519638271495, −8.423022581130290576301991169454, −7.49882222813615492496690954815, −6.09718619139348092858449290334, −5.11444113206155616914735686192, −3.76814264513886101656957905129, −2.73158671569481376590661140528, 1.65347948816690566073220743125, 3.04970915629806773856084623272, 4.18362043781432014364862887375, 5.93408579055815809346835975376, 6.81523098602814247272766719362, 7.928644980828661607196671316671, 9.441501336624796392915448231165, 9.693557773998500056471481753142, 11.32904005683097425893185408272, 12.04632660220044782760304229245

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