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Properties

Label 2-234-13.9-c1-0-1
Degree $2$
Conductor $234$
Sign $0.522 - 0.852i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 5-s + (1 + 1.73i)7-s + 0.999·8-s + (−0.5 + 0.866i)10-s + (1 − 1.73i)11-s + (2.5 + 2.59i)13-s − 1.99·14-s + (−0.5 + 0.866i)16-s + (2.5 + 4.33i)17-s + (1 + 1.73i)19-s + (−0.499 − 0.866i)20-s + (0.999 + 1.73i)22-s + (3 − 5.19i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (0.377 + 0.654i)7-s + 0.353·8-s + (−0.158 + 0.273i)10-s + (0.301 − 0.522i)11-s + (0.693 + 0.720i)13-s − 0.534·14-s + (−0.125 + 0.216i)16-s + (0.606 + 1.05i)17-s + (0.229 + 0.397i)19-s + (−0.111 − 0.193i)20-s + (0.213 + 0.369i)22-s + (0.625 − 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.522 - 0.852i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00424 + 0.562754i\)
\(L(\frac12)\) \(\approx\) \(1.00424 + 0.562754i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
13 \( 1 + (-2.5 - 2.59i)T \)
good5 \( 1 - T + 5T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - T + 53T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7 + 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.36980332754518205383837820680, −11.22450371136240726359428729421, −10.35497310101520882453885529515, −9.064873862233129579461606787888, −8.623069487961979142990034376591, −7.35444231705528095951801077106, −6.12477808912279221225894327054, −5.44999708650564097041351758657, −3.82432642323044593175718448399, −1.76866998700126560472139144312, 1.32891609226434304170674118202, 3.06504907969230251247033175902, 4.44444134367803266578672192696, 5.76071098180468240682513108642, 7.27302869645441159250072054504, 8.062722273869621878422568846984, 9.492123706298210017363199593872, 9.915061611571395512772921085046, 11.20055248403727682303013496921, 11.68144190921400929482720113246

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