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Properties

Label 2-234-117.103-c1-0-8
Degree $2$
Conductor $234$
Sign $0.998 - 0.0628i$
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 + (0.523 + 1.65i)3-s + (0.499 − 0.866i)4-s + (−0.419 − 0.242i)5-s + (1.27 + 1.16i)6-s + (4.37 − 2.52i)7-s − 0.999i·8-s + (−2.45 + 1.72i)9-s − 0.484·10-s + (−2.78 + 1.60i)11-s + (1.69 + 0.372i)12-s + (−0.722 + 3.53i)13-s + (2.52 − 4.37i)14-s + (0.180 − 0.819i)15-s + (−0.5 − 0.866i)16-s + 4.20·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.302 + 0.953i)3-s + (0.249 − 0.433i)4-s + (−0.187 − 0.108i)5-s + (0.521 + 0.476i)6-s + (1.65 − 0.955i)7-s − 0.353i·8-s + (−0.817 + 0.575i)9-s − 0.153·10-s + (−0.838 + 0.484i)11-s + (0.488 + 0.107i)12-s + (−0.200 + 0.979i)13-s + (0.675 − 1.16i)14-s + (0.0465 − 0.211i)15-s + (−0.125 − 0.216i)16-s + 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.998 - 0.0628i$
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: \((2,\ 234,\ (\ :1/2),\ 0.998 - 0.0628i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92307 + 0.0604946i\)
\(L(\frac12)\) \(\approx\) \(1.92307 + 0.0604946i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.523 - 1.65i)T \)
13 \( 1 + (0.722 - 3.53i)T \)
good5 \( 1 + (0.419 + 0.242i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-4.37 + 2.52i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.78 - 1.60i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 4.20T + 17T^{2} \)
19 \( 1 + 3.21iT - 19T^{2} \)
23 \( 1 + (3.13 - 5.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.29 + 3.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.61 + 3.24i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.08iT - 37T^{2} \)
41 \( 1 + (9.57 + 5.52i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.73 - 8.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.57 - 2.64i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.41T + 53T^{2} \)
59 \( 1 + (-3.13 - 1.81i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.500 - 0.867i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.936 - 0.540i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.63iT - 71T^{2} \)
73 \( 1 - 0.325iT - 73T^{2} \)
79 \( 1 + (-3.91 - 6.78i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.08 - 2.93i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.42iT - 89T^{2} \)
97 \( 1 + (-11.3 + 6.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.83908769594363061288719182794, −11.30445126111200316527822951344, −10.38788890916419891068670479641, −9.582225356780364229103601943021, −8.139444100513985447808909912517, −7.41795674966070994729076327883, −5.45805951283634357084208518174, −4.62317197358608319688315251680, −3.84797188711701029667793640264, −2.05011077504871460122076348568, 1.95972902279580151126550800761, 3.30630974924463602944335801523, 5.26295149731319636375822914671, 5.72883437381789538563360127431, 7.38140907671964469391248692635, 8.065947964994334408109876822105, 8.637791747780199068349178443578, 10.53446899025008018972216725863, 11.58509894114505109941267893331, 12.27444030076020764473218228433

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