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Properties

Label 2-234-117.103-c1-0-5
Degree $2$
Conductor $234$
Sign $0.728 + 0.684i$
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  + (−0.866 + 0.5i)2-s + (−0.579 + 1.63i)3-s + (0.499 − 0.866i)4-s + (−3.32 − 1.91i)5-s + (−0.314 − 1.70i)6-s + (2.91 − 1.68i)7-s + 0.999i·8-s + (−2.32 − 1.89i)9-s + 3.83·10-s + (1.72 − 0.995i)11-s + (1.12 + 1.31i)12-s + (−0.985 − 3.46i)13-s + (−1.68 + 2.91i)14-s + (5.05 − 4.31i)15-s + (−0.5 − 0.866i)16-s + 2.60·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.334 + 0.942i)3-s + (0.249 − 0.433i)4-s + (−1.48 − 0.857i)5-s + (−0.128 − 0.695i)6-s + (1.10 − 0.636i)7-s + 0.353i·8-s + (−0.776 − 0.630i)9-s + 1.21·10-s + (0.520 − 0.300i)11-s + (0.324 + 0.380i)12-s + (−0.273 − 0.961i)13-s + (−0.450 + 0.779i)14-s + (1.30 − 1.11i)15-s + (−0.125 − 0.216i)16-s + 0.630·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.728 + 0.684i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.728 + 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.556911 - 0.220660i\)
\(L(\frac12)\) \(\approx\) \(0.556911 - 0.220660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.579 - 1.63i)T \)
13 \( 1 + (0.985 + 3.46i)T \)
good5 \( 1 + (3.32 + 1.91i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.91 + 1.68i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.72 + 0.995i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.60T + 17T^{2} \)
19 \( 1 - 1.99iT - 19T^{2} \)
23 \( 1 + (-2.13 + 3.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.37 + 7.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.57 + 2.64i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.08iT - 37T^{2} \)
41 \( 1 + (-7.13 - 4.12i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.13 + 8.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.22 - 2.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.16T + 53T^{2} \)
59 \( 1 + (-6.33 - 3.65i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.63 - 9.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.90 + 2.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 - 8.41iT - 73T^{2} \)
79 \( 1 + (-2.97 - 5.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.03 + 1.17i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.28iT - 89T^{2} \)
97 \( 1 + (-8.92 + 5.15i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.63255736193953416874582531761, −11.16516061912183657198291421642, −10.16479966413188274683725354881, −8.975295680086208942474541887875, −8.115674711576626579463557744032, −7.50626569140115857001132887499, −5.68583462026333758853900065066, −4.63925065257298138073925738083, −3.74742622915692270406001412105, −0.66017336061119900310996449685, 1.73255538269655530829488288100, 3.31602315052713877078848801825, 4.93796269094145841954998369654, 6.70358136859963011518104733546, 7.42519876872847634014647395313, 8.145465452074967399843340879825, 9.179085059470702808828981315201, 10.88546067082397737785966240846, 11.46805850620971836630567555421, 11.85375008041248590083250151878

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