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Properties

Label 2-234-117.103-c1-0-13
Degree $2$
Conductor $234$
Sign $-0.781 + 0.624i$
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 + (−0.987 − 1.42i)3-s + (0.499 − 0.866i)4-s + (−2.53 − 1.46i)5-s + (−1.56 − 0.738i)6-s + (1.90 − 1.10i)7-s − 0.999i·8-s + (−1.05 + 2.81i)9-s − 2.93·10-s + (−4.47 + 2.58i)11-s + (−1.72 + 0.143i)12-s + (0.680 − 3.54i)13-s + (1.10 − 1.90i)14-s + (0.420 + 5.06i)15-s + (−0.5 − 0.866i)16-s + 2.31·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.569 − 0.821i)3-s + (0.249 − 0.433i)4-s + (−1.13 − 0.655i)5-s + (−0.639 − 0.301i)6-s + (0.721 − 0.416i)7-s − 0.353i·8-s + (−0.350 + 0.936i)9-s − 0.927·10-s + (−1.34 + 0.779i)11-s + (−0.498 + 0.0414i)12-s + (0.188 − 0.982i)13-s + (0.294 − 0.510i)14-s + (0.108 + 1.30i)15-s + (−0.125 − 0.216i)16-s + 0.560·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.781 + 0.624i$
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: \((2,\ 234,\ (\ :1/2),\ -0.781 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.366180 - 1.04525i\)
\(L(\frac12)\) \(\approx\) \(0.366180 - 1.04525i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.987 + 1.42i)T \)
13 \( 1 + (-0.680 + 3.54i)T \)
good5 \( 1 + (2.53 + 1.46i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.90 + 1.10i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.47 - 2.58i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.31T + 17T^{2} \)
19 \( 1 + 5.16iT - 19T^{2} \)
23 \( 1 + (-4.19 + 7.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.72 - 8.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.38 - 3.11i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.646iT - 37T^{2} \)
41 \( 1 + (-0.674 - 0.389i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.74 + 3.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.79 + 2.76i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.68T + 53T^{2} \)
59 \( 1 + (2.59 + 1.49i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.432 - 0.748i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.68 - 5.58i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 - 4.27iT - 73T^{2} \)
79 \( 1 + (-1.52 - 2.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.19 + 1.84i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.34iT - 89T^{2} \)
97 \( 1 + (-1.29 + 0.745i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.10771676733922705953716998127, −10.91628010958709628494528092240, −10.51280322888548408589233521577, −8.453355183333281478434747251519, −7.75702255968205568062323138303, −6.81716675094813077234132803043, −5.05051313545968453046031266031, −4.77723812719444748952852860865, −2.81025934689809802631015151265, −0.834230367670584940532203076642, 3.09442442470589981293923557977, 4.14374273531240977433326481568, 5.25086675194155162591555435121, 6.20417586682641261071415160390, 7.64600145321505066081227787226, 8.318953532791950535844929083914, 9.850737832524678047849788999484, 11.07795918388571672712030550761, 11.46506583561583680615377892372, 12.23192057200073755777766290772

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