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Properties

Label 2-234-117.20-c1-0-13
Degree $2$
Conductor $234$
Sign $0.495 + 0.868i$
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  + (0.965 + 0.258i)2-s + (0.161 − 1.72i)3-s + (0.866 + 0.499i)4-s + (−2.21 − 0.593i)5-s + (0.602 − 1.62i)6-s + (2.94 − 2.94i)7-s + (0.707 + 0.707i)8-s + (−2.94 − 0.556i)9-s + (−1.98 − 1.14i)10-s + (−0.431 + 1.60i)11-s + (1.00 − 1.41i)12-s + (1.58 − 3.23i)13-s + (3.60 − 2.08i)14-s + (−1.38 + 3.72i)15-s + (0.500 + 0.866i)16-s + (1.96 + 3.41i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.0932 − 0.995i)3-s + (0.433 + 0.249i)4-s + (−0.990 − 0.265i)5-s + (0.245 − 0.662i)6-s + (1.11 − 1.11i)7-s + (0.249 + 0.249i)8-s + (−0.982 − 0.185i)9-s + (−0.628 − 0.362i)10-s + (−0.129 + 0.485i)11-s + (0.289 − 0.407i)12-s + (0.439 − 0.898i)13-s + (0.964 − 0.556i)14-s + (−0.356 + 0.961i)15-s + (0.125 + 0.216i)16-s + (0.477 + 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.495 + 0.868i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.495 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47800 - 0.858484i\)
\(L(\frac12)\) \(\approx\) \(1.47800 - 0.858484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-0.161 + 1.72i)T \)
13 \( 1 + (-1.58 + 3.23i)T \)
good5 \( 1 + (2.21 + 0.593i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.94 + 2.94i)T - 7iT^{2} \)
11 \( 1 + (0.431 - 1.60i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.96 - 3.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.878 - 3.27i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 5.07T + 23T^{2} \)
29 \( 1 + (4.53 - 2.61i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.610 - 2.27i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.21 - 4.55i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.88 + 2.88i)T - 41iT^{2} \)
43 \( 1 - 4.92iT - 43T^{2} \)
47 \( 1 + (-4.39 + 1.17i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 6.69iT - 53T^{2} \)
59 \( 1 + (8.36 - 2.24i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + (9.81 + 9.81i)T + 67iT^{2} \)
71 \( 1 + (13.5 + 3.63i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.61 + 6.61i)T - 73iT^{2} \)
79 \( 1 + (8.13 - 14.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.02 + 7.55i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-8.95 + 2.40i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.05 + 6.05i)T + 97iT^{2} \)
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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

−12.25630363429544012397180794921, −11.25197850467177130254035402577, −10.56094736053001621615739777649, −8.502753015255585506027179553604, −7.75597279842494361815538367693, −7.28108643884573526483019671867, −5.82843035570733005325632897713, −4.55837439515377923400441498286, −3.42564655659896838692335931939, −1.38116461148379341500836611215, 2.58370289961841285962198997534, 3.85997956113235884820837962234, 4.86621769724656660292531880242, 5.77098102103636828503764268694, 7.40550295390405902315908570520, 8.555422650214004070370147074693, 9.341779654041291949835848556995, 10.92484104372173002773331024154, 11.42877164679856774867327121856, 11.87468029941098504461278393073

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