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Properties

Label 2-234-117.11-c1-0-12
Degree $2$
Conductor $234$
Sign $-0.972 + 0.234i$
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 + (−1.36 + 1.06i)3-s − 1.00i·4-s + (−3.36 − 0.901i)5-s + (−0.211 + 1.71i)6-s + (0.0541 + 0.0145i)7-s + (−0.707 − 0.707i)8-s + (0.727 − 2.91i)9-s + (−3.01 + 1.74i)10-s + (−3.65 − 3.65i)11-s + (1.06 + 1.36i)12-s + (−3.37 + 1.25i)13-s + (0.0485 − 0.0280i)14-s + (5.55 − 2.35i)15-s − 1.00·16-s + (−1.83 + 3.17i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.788 + 0.615i)3-s − 0.500i·4-s + (−1.50 − 0.403i)5-s + (−0.0863 + 0.701i)6-s + (0.0204 + 0.00548i)7-s + (−0.250 − 0.250i)8-s + (0.242 − 0.970i)9-s + (−0.954 + 0.550i)10-s + (−1.10 − 1.10i)11-s + (0.307 + 0.394i)12-s + (−0.937 + 0.348i)13-s + (0.0129 − 0.00749i)14-s + (1.43 − 0.608i)15-s − 0.250·16-s + (−0.443 + 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.972 + 0.234i$
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: \((2,\ 234,\ (\ :1/2),\ -0.972 + 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0421597 - 0.354394i\)
\(L(\frac12)\) \(\approx\) \(0.0421597 - 0.354394i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.36 - 1.06i)T \)
13 \( 1 + (3.37 - 1.25i)T \)
good5 \( 1 + (3.36 + 0.901i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.0541 - 0.0145i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.65 + 3.65i)T + 11iT^{2} \)
17 \( 1 + (1.83 - 3.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.677 + 0.181i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.89 + 5.01i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.45iT - 29T^{2} \)
31 \( 1 + (-1.54 + 5.75i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-7.28 - 1.95i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.780 + 2.91i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.70 - 2.71i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.82 - 1.29i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 3.89iT - 53T^{2} \)
59 \( 1 + (-0.745 - 0.745i)T + 59iT^{2} \)
61 \( 1 + (-1.50 - 2.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.5 - 2.81i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.98 + 14.8i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.86 + 5.86i)T - 73iT^{2} \)
79 \( 1 + (-2.76 + 4.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.13 + 15.4i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (4.41 - 16.4i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.595 - 2.22i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.60516618092853872925255285108, −11.06647162430636886276903175600, −10.19161635545162009487479391910, −8.855983949840098493318263809724, −7.80454270475438437600529317303, −6.39003151798883363574597819336, −5.06119630004799371695619801222, −4.35258276183072364261775650455, −3.15964905747615660937125753374, −0.26178179756609240541533261293, 2.79390830393581665291141121950, 4.50979577668567553296086977940, 5.25345625128837565431112852757, 6.86635007375612208206516326800, 7.45157781708551365029225632341, 8.045576306464928065793494564088, 9.905959479798266203450587556796, 11.11881057815362785423547240306, 11.78485630885711604546190863360, 12.56424586291609200227499125652

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