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Properties

Label 2-234-117.11-c1-0-11
Degree $2$
Conductor $234$
Sign $-0.398 + 0.917i$
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  + (−0.707 + 0.707i)2-s + (1.46 − 0.920i)3-s − 1.00i·4-s + (−3.62 − 0.970i)5-s + (−0.386 + 1.68i)6-s + (−3.13 − 0.840i)7-s + (0.707 + 0.707i)8-s + (1.30 − 2.70i)9-s + (3.24 − 1.87i)10-s + (−1.54 − 1.54i)11-s + (−0.920 − 1.46i)12-s + (0.305 − 3.59i)13-s + (2.81 − 1.62i)14-s + (−6.20 + 1.90i)15-s − 1.00·16-s + (−3.50 + 6.07i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.847 − 0.531i)3-s − 0.500i·4-s + (−1.61 − 0.433i)5-s + (−0.157 + 0.689i)6-s + (−1.18 − 0.317i)7-s + (0.250 + 0.250i)8-s + (0.435 − 0.900i)9-s + (1.02 − 0.592i)10-s + (−0.466 − 0.466i)11-s + (−0.265 − 0.423i)12-s + (0.0847 − 0.996i)13-s + (0.751 − 0.434i)14-s + (−1.60 + 0.493i)15-s − 0.250·16-s + (−0.850 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.398 + 0.917i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.398 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.323588 - 0.493123i\)
\(L(\frac12)\) \(\approx\) \(0.323588 - 0.493123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-1.46 + 0.920i)T \)
13 \( 1 + (-0.305 + 3.59i)T \)
good5 \( 1 + (3.62 + 0.970i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (3.13 + 0.840i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.54 + 1.54i)T + 11iT^{2} \)
17 \( 1 + (3.50 - 6.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.70 + 1.26i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.415 + 0.720i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.26iT - 29T^{2} \)
31 \( 1 + (0.141 - 0.527i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.90 - 0.509i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.455 - 1.70i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.0503 + 0.0290i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.83 + 1.56i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 1.62iT - 53T^{2} \)
59 \( 1 + (-2.06 - 2.06i)T + 59iT^{2} \)
61 \( 1 + (1.78 + 3.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.5 - 3.36i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.31 + 8.65i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.38 + 3.38i)T - 73iT^{2} \)
79 \( 1 + (5.86 - 10.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.75 - 14.0i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-3.50 + 13.0i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.55 + 9.53i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.03074364160026922251905871478, −10.80415968259467608009502100858, −9.684016369733334097313495479945, −8.562019274508685340554084101128, −7.986918861537168558575839604106, −7.20082605575935112959413979843, −6.04954013551769667991339416734, −4.14342575215319929929669250670, −3.10403998994392163057218815516, −0.50446922619330655031139896254, 2.74427461422090755103101498769, 3.54607889338857463233865418916, 4.67711730111268291149035184041, 7.03786007603023722916131678376, 7.52947320134811740993530757409, 8.867220014798975447830145173163, 9.411651427954764093860918296860, 10.50932840702392159635766804887, 11.46235526604682834195138875459, 12.24099644076468710177521130630

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