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Properties

Label 2-234-117.110-c1-0-0
Degree $2$
Conductor $234$
Sign $-0.954 + 0.298i$
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  + (−0.258 + 0.965i)2-s + (−0.788 − 1.54i)3-s + (−0.866 − 0.499i)4-s + (−0.707 + 2.64i)5-s + (1.69 − 0.362i)6-s + (−2.46 − 2.46i)7-s + (0.707 − 0.707i)8-s + (−1.75 + 2.43i)9-s + (−2.36 − 1.36i)10-s + (−1.76 − 0.472i)11-s + (−0.0878 + 1.72i)12-s + (−3.37 + 1.25i)13-s + (3.01 − 1.74i)14-s + (4.63 − 0.992i)15-s + (0.500 + 0.866i)16-s + (0.419 + 0.726i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.455 − 0.890i)3-s + (−0.433 − 0.249i)4-s + (−0.316 + 1.18i)5-s + (0.691 − 0.148i)6-s + (−0.931 − 0.931i)7-s + (0.249 − 0.249i)8-s + (−0.585 + 0.810i)9-s + (−0.748 − 0.432i)10-s + (−0.531 − 0.142i)11-s + (−0.0253 + 0.499i)12-s + (−0.937 + 0.348i)13-s + (0.806 − 0.465i)14-s + (1.19 − 0.256i)15-s + (0.125 + 0.216i)16-s + (0.101 + 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.954 + 0.298i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.954 + 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00343568 - 0.0225034i\)
\(L(\frac12)\) \(\approx\) \(0.00343568 - 0.0225034i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.788 + 1.54i)T \)
13 \( 1 + (3.37 - 1.25i)T \)
good5 \( 1 + (0.707 - 2.64i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.46 + 2.46i)T + 7iT^{2} \)
11 \( 1 + (1.76 + 0.472i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.419 - 0.726i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.94 + 1.86i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 6.37T + 23T^{2} \)
29 \( 1 + (-5.24 + 3.02i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.43 - 1.99i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.09 - 1.09i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.93 - 2.93i)T + 41iT^{2} \)
43 \( 1 + 8.62iT - 43T^{2} \)
47 \( 1 + (-2.00 - 7.46i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 3.92iT - 53T^{2} \)
59 \( 1 + (-1.92 - 7.19i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 0.494T + 61T^{2} \)
67 \( 1 + (3.91 - 3.91i)T - 67iT^{2} \)
71 \( 1 + (1.48 - 5.55i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.66 + 9.66i)T + 73iT^{2} \)
79 \( 1 + (0.892 - 1.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (15.2 - 4.07i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.385 + 1.43i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.2 + 12.2i)T - 97iT^{2} \)
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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

−12.80553776200676464311421893182, −11.80142265556088659466751889784, −10.51487931134951506357662622226, −10.16199919469238777129514213740, −8.403160824553208146073384570373, −7.39882297789909647750055940341, −6.76690767114782888545021900381, −6.09352722129867351107972160896, −4.36789008482486142457828545809, −2.66820454735405942825385150268, 0.01938999278859561199573278094, 2.67859133338847134674517494107, 4.19312901811426896291008297674, 5.09611356325495015869903879509, 6.20057679467723094288627685124, 8.184535659628637979845564698267, 8.919995855636945751944857778176, 9.859118955533875746015365086179, 10.45892067426076768064752467219, 11.95547822422706772793752074994

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