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Properties

Label 2-234-117.16-c1-0-8
Degree $2$
Conductor $234$
Sign $0.945 + 0.324i$
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  + 2-s + (1.15 − 1.28i)3-s + 4-s + (0.635 + 1.10i)5-s + (1.15 − 1.28i)6-s + (0.919 + 1.59i)7-s + 8-s + (−0.320 − 2.98i)9-s + (0.635 + 1.10i)10-s − 4.98·11-s + (1.15 − 1.28i)12-s + (−3.13 + 1.77i)13-s + (0.919 + 1.59i)14-s + (2.15 + 0.455i)15-s + 16-s + (1.76 − 3.05i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.668 − 0.743i)3-s + 0.5·4-s + (0.284 + 0.491i)5-s + (0.472 − 0.526i)6-s + (0.347 + 0.601i)7-s + 0.353·8-s + (−0.106 − 0.994i)9-s + (0.200 + 0.347i)10-s − 1.50·11-s + (0.334 − 0.371i)12-s + (−0.869 + 0.493i)13-s + (0.245 + 0.425i)14-s + (0.555 + 0.117i)15-s + 0.250·16-s + (0.428 − 0.742i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.945 + 0.324i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.945 + 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14487 - 0.358080i\)
\(L(\frac12)\) \(\approx\) \(2.14487 - 0.358080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.15 + 1.28i)T \)
13 \( 1 + (3.13 - 1.77i)T \)
good5 \( 1 + (-0.635 - 1.10i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.919 - 1.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.98T + 11T^{2} \)
17 \( 1 + (-1.76 + 3.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.245 + 0.425i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0529 - 0.0916i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.30T + 29T^{2} \)
31 \( 1 + (-2.94 - 5.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.354 - 0.614i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.36 + 4.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.05 - 7.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.67 - 8.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.00T + 53T^{2} \)
59 \( 1 + 9.54T + 59T^{2} \)
61 \( 1 + (-3.17 - 5.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.00 - 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.142 + 0.247i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + (-4.96 + 8.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.39 + 11.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.30 + 10.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.37 - 5.84i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.35058638587843551553899449886, −11.45358960152833103999225621843, −10.26493164322908074529248697713, −9.146525037976405542750923464156, −7.86186997366274104022608275753, −7.18878634168513406911658651717, −5.95836074617115160192178873568, −4.85680833504755092033742440760, −3.00418869106455651714967268218, −2.22781480644899678244644702508, 2.30679655971293196883447090901, 3.67174470307051232043832833903, 4.89557452949835159781782852590, 5.56057419009630190520926949694, 7.50736257328787907519442033823, 8.083751189955847979507390322748, 9.497236617365604676467011567548, 10.37377733640459968149152465970, 11.06969470894128532029643215650, 12.57871575051639295641298994695

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