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Properties

Label 2-234-117.41-c1-0-2
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: $\chi_{234} (41, \cdot )$
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

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

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