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Properties

Label 2-234-117.103-c1-0-11
Degree $2$
Conductor $234$
Sign $-0.998 - 0.0499i$
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 + (−0.215 − 1.71i)3-s + (0.499 − 0.866i)4-s + (−2.40 − 1.38i)5-s + (1.04 + 1.38i)6-s + (−0.759 + 0.438i)7-s + 0.999i·8-s + (−2.90 + 0.739i)9-s + 2.77·10-s + (−1.92 + 1.11i)11-s + (−1.59 − 0.672i)12-s + (0.180 + 3.60i)13-s + (0.438 − 0.759i)14-s + (−1.86 + 4.43i)15-s + (−0.5 − 0.866i)16-s − 3.80·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.124 − 0.992i)3-s + (0.249 − 0.433i)4-s + (−1.07 − 0.621i)5-s + (0.426 + 0.563i)6-s + (−0.286 + 0.165i)7-s + 0.353i·8-s + (−0.969 + 0.246i)9-s + 0.878·10-s + (−0.581 + 0.335i)11-s + (−0.460 − 0.194i)12-s + (0.0499 + 0.998i)13-s + (0.117 − 0.202i)14-s + (−0.482 + 1.14i)15-s + (−0.125 − 0.216i)16-s − 0.923·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.998 - 0.0499i$
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: \((2,\ 234,\ (\ :1/2),\ -0.998 - 0.0499i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00459662 + 0.184106i\)
\(L(\frac12)\) \(\approx\) \(0.00459662 + 0.184106i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.215 + 1.71i)T \)
13 \( 1 + (-0.180 - 3.60i)T \)
good5 \( 1 + (2.40 + 1.38i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.759 - 0.438i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.92 - 1.11i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.80T + 17T^{2} \)
19 \( 1 + 2.22iT - 19T^{2} \)
23 \( 1 + (-0.259 + 0.449i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.81 + 6.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.97 + 2.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.3iT - 37T^{2} \)
41 \( 1 + (3.52 + 2.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.81 - 4.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.920 + 0.531i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.29T + 53T^{2} \)
59 \( 1 + (-3.52 - 2.03i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.94 + 6.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.95 - 3.43i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.9iT - 71T^{2} \)
73 \( 1 + 5.24iT - 73T^{2} \)
79 \( 1 + (7.09 + 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.641 - 0.370i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.89iT - 89T^{2} \)
97 \( 1 + (13.6 - 7.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.62680740226620312902661429905, −11.01780376987087781608488538813, −9.376015341933885465243554856698, −8.597116239560738905760335011207, −7.64391275421923405910319469033, −6.95305833587971595810862731125, −5.72675993555264815172923726517, −4.28576477999555865103491713814, −2.20206854333585498078501237327, −0.17479350460134072723781290997, 3.02627355683493464037854869151, 3.79769189613992710130313349386, 5.30692541755302680743277555490, 6.83421081267066051515075865832, 7.975260229757785515311372105147, 8.788940066070721758208790775687, 10.02620653277006528822256853189, 10.74994293600926345842159691503, 11.29110203307897059270011410484, 12.33231219484803472913442046001

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