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Properties

Label 2-234-117.11-c1-0-6
Degree $2$
Conductor $234$
Sign $0.999 + 0.00318i$
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 + (1.04 + 1.38i)3-s − 1.00i·4-s + (1.62 + 0.435i)5-s + (1.71 + 0.239i)6-s + (0.290 + 0.0778i)7-s + (−0.707 − 0.707i)8-s + (−0.823 + 2.88i)9-s + (1.45 − 0.842i)10-s + (−1.12 − 1.12i)11-s + (1.38 − 1.04i)12-s + (−3.56 + 0.535i)13-s + (0.260 − 0.150i)14-s + (1.09 + 2.70i)15-s − 1.00·16-s + (1.26 − 2.19i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.602 + 0.798i)3-s − 0.500i·4-s + (0.727 + 0.194i)5-s + (0.700 + 0.0979i)6-s + (0.109 + 0.0294i)7-s + (−0.250 − 0.250i)8-s + (−0.274 + 0.961i)9-s + (0.461 − 0.266i)10-s + (−0.338 − 0.338i)11-s + (0.399 − 0.301i)12-s + (−0.988 + 0.148i)13-s + (0.0695 − 0.0401i)14-s + (0.282 + 0.698i)15-s − 0.250·16-s + (0.306 − 0.531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.999 + 0.00318i$
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: \((2,\ 234,\ (\ :1/2),\ 0.999 + 0.00318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96814 - 0.00313531i\)
\(L(\frac12)\) \(\approx\) \(1.96814 - 0.00313531i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 - 1.38i)T \)
13 \( 1 + (3.56 - 0.535i)T \)
good5 \( 1 + (-1.62 - 0.435i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.290 - 0.0778i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.12 + 1.12i)T + 11iT^{2} \)
17 \( 1 + (-1.26 + 2.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.16 + 1.11i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.660 + 1.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.48iT - 29T^{2} \)
31 \( 1 + (2.78 - 10.3i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.37 + 0.637i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.974 + 3.63i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.42 - 3.71i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.46 + 0.928i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 3.63iT - 53T^{2} \)
59 \( 1 + (0.512 + 0.512i)T + 59iT^{2} \)
61 \( 1 + (-5.62 - 9.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.943 + 0.252i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.84 - 6.87i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.38 + 3.38i)T - 73iT^{2} \)
79 \( 1 + (3.69 - 6.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.18 - 11.8i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-0.512 + 1.91i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.61 + 6.03i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.11926803777674997458921961354, −11.15418307685596137797455191491, −10.05838192456401830355379886850, −9.684760633555167842728248708199, −8.484453771274822053027588808554, −7.14812636656392350548905432744, −5.57199187672814893662138666699, −4.79740141989474723611141602081, −3.32405194246256856200931135975, −2.26923340773432529639157721304, 1.94937650911063441004162488364, 3.37275696966261501684719689553, 5.07510136385105224678174325053, 6.05335940384379495364720069256, 7.26964467323812310588730079212, 7.907770267205417785828651647595, 9.162473658565570833446347925999, 9.985529000715378818356131594002, 11.58330786422956916931153828857, 12.55344430245041206394042988818

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