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Properties

Label 2-234-117.103-c1-0-1
Degree $2$
Conductor $234$
Sign $-0.835 - 0.548i$
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 + (−2.69 + 2.39i)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 + (−0.748 + 0.663i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.835 - 0.548i$
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: \((2,\ 234,\ (\ :1/2),\ -0.835 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.217488 + 0.727609i\)
\(L(\frac12)\) \(\approx\) \(0.217488 + 0.727609i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.69 - 2.39i)T \)
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

−12.31886349323643690850849151730, −11.57362580706695059872060581876, −10.04993030058563930795884076208, −9.671135368425318772949203821685, −9.007673480722066334742770541184, −7.81606301466223991105650757372, −6.28848727969463874937891352717, −5.70594670386442350599872392075, −3.89186922229373098476020668986, −2.63738366122715544227832534668, 0.71931663584770942631582219792, 2.62164727764586318473373408956, 3.80952257636989411526864995036, 6.01254117880649754959335792000, 7.02599497772931341253132404473, 7.60977773957647379740179390616, 9.060963319071470062423008746497, 9.735832161920703313303338055026, 10.62095862390008449431059041563, 12.12540120672950919225836110084

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