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Properties

Label 2-234-117.103-c1-0-7
Degree $2$
Conductor $234$
Sign $0.679 + 0.733i$
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 + (−1.62 + 0.612i)3-s + (0.499 − 0.866i)4-s + (−0.548 − 0.316i)5-s + (−1.09 + 1.34i)6-s + (2.15 − 1.24i)7-s − 0.999i·8-s + (2.24 − 1.98i)9-s − 0.633·10-s + (4.20 − 2.43i)11-s + (−0.279 + 1.70i)12-s + (3.35 − 1.31i)13-s + (1.24 − 2.15i)14-s + (1.08 + 0.176i)15-s + (−0.5 − 0.866i)16-s − 6.27·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.935 + 0.353i)3-s + (0.249 − 0.433i)4-s + (−0.245 − 0.141i)5-s + (−0.447 + 0.547i)6-s + (0.814 − 0.469i)7-s − 0.353i·8-s + (0.749 − 0.661i)9-s − 0.200·10-s + (1.26 − 0.732i)11-s + (−0.0806 + 0.493i)12-s + (0.931 − 0.364i)13-s + (0.332 − 0.575i)14-s + (0.279 + 0.0456i)15-s + (−0.125 − 0.216i)16-s − 1.52·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.679 + 0.733i$
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: \((2,\ 234,\ (\ :1/2),\ 0.679 + 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28275 - 0.560625i\)
\(L(\frac12)\) \(\approx\) \(1.28275 - 0.560625i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 0.612i)T \)
13 \( 1 + (-3.35 + 1.31i)T \)
good5 \( 1 + (0.548 + 0.316i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.15 + 1.24i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.20 + 2.43i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 6.27T + 17T^{2} \)
19 \( 1 - 4.86iT - 19T^{2} \)
23 \( 1 + (1.77 - 3.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.415 - 0.719i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.73 - 2.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.81iT - 37T^{2} \)
41 \( 1 + (0.0678 + 0.0391i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.84 - 8.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.30 - 2.48i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.36T + 53T^{2} \)
59 \( 1 + (7.86 + 4.54i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.28 - 9.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.82 - 3.94i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 + 1.05iT - 73T^{2} \)
79 \( 1 + (1.68 + 2.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-13.1 + 7.60i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.595iT - 89T^{2} \)
97 \( 1 + (14.7 - 8.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.80420830359170090195569012047, −11.22947212151105804403541738070, −10.59695927002845093845215604757, −9.325835833000865689858877829587, −8.091987151532828361019424527845, −6.56695298568894341890379929679, −5.83022482808729063705708569896, −4.44281190389228899200570267702, −3.80152636221526684637202932210, −1.30593916636605420576109551299, 1.91673098150485327508513383684, 4.16392913580675182374871752687, 4.94386805429219411115670449567, 6.36704664351524995309076567439, 6.84503974445422516359062644670, 8.167687624231232244730074048693, 9.272263804596356146678811054692, 10.93310903129930705146251852921, 11.50100965579307606537076619764, 12.13017409684477920304956256352

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