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Properties

Label 2-234-117.110-c1-0-4
Degree $2$
Conductor $234$
Sign $0.593 - 0.804i$
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 + (−1.66 − 0.491i)3-s + (−0.866 − 0.499i)4-s + (0.174 − 0.650i)5-s + (0.904 − 1.47i)6-s + (1.26 + 1.26i)7-s + (0.707 − 0.707i)8-s + (2.51 + 1.63i)9-s + (0.583 + 0.336i)10-s + (0.952 + 0.255i)11-s + (1.19 + 1.25i)12-s + (2.90 + 2.13i)13-s + (−1.55 + 0.895i)14-s + (−0.609 + 0.995i)15-s + (0.500 + 0.866i)16-s + (1.20 + 2.09i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.958 − 0.283i)3-s + (−0.433 − 0.249i)4-s + (0.0779 − 0.290i)5-s + (0.369 − 0.603i)6-s + (0.478 + 0.478i)7-s + (0.249 − 0.249i)8-s + (0.839 + 0.543i)9-s + (0.184 + 0.106i)10-s + (0.287 + 0.0769i)11-s + (0.344 + 0.362i)12-s + (0.806 + 0.591i)13-s + (−0.414 + 0.239i)14-s + (−0.157 + 0.256i)15-s + (0.125 + 0.216i)16-s + (0.293 + 0.508i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.593 - 0.804i$
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: \((2,\ 234,\ (\ :1/2),\ 0.593 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.801856 + 0.405087i\)
\(L(\frac12)\) \(\approx\) \(0.801856 + 0.405087i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 0.491i)T \)
13 \( 1 + (-2.90 - 2.13i)T \)
good5 \( 1 + (-0.174 + 0.650i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.26 - 1.26i)T + 7iT^{2} \)
11 \( 1 + (-0.952 - 0.255i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.20 - 2.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.33 - 1.69i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 6.32T + 23T^{2} \)
29 \( 1 + (-3.71 + 2.14i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.59 + 1.49i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.83 + 1.83i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.51 + 3.51i)T + 41iT^{2} \)
43 \( 1 - 0.892iT - 43T^{2} \)
47 \( 1 + (0.981 + 3.66i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 9.00iT - 53T^{2} \)
59 \( 1 + (-1.22 - 4.56i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 4.76T + 61T^{2} \)
67 \( 1 + (7.82 - 7.82i)T - 67iT^{2} \)
71 \( 1 + (-1.70 + 6.37i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (8.01 + 8.01i)T + 73iT^{2} \)
79 \( 1 + (0.807 - 1.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.31 - 1.95i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.440 - 1.64i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.35 - 1.35i)T - 97iT^{2} \)
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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

−12.16276139733076470949349985335, −11.52714693254410757419349035733, −10.38653234894539945573550586364, −9.327364027241986038781138189512, −8.236723517809699604212835029662, −7.24929569940559141662162595934, −6.07595436049210569931815717928, −5.40547813566063551618744450199, −4.13696287235694347017417079892, −1.46219059659972114719708520531, 1.11138714411019042341064930272, 3.30793063099608347642483748405, 4.57556492811170718819440203439, 5.69224037331412781789835311600, 6.97113131793891625067810721938, 8.136606244421690450385546227827, 9.513436405172821321300579755394, 10.29561147637493068926138202383, 11.13509101413752679999125516095, 11.72574862336085423533689749941

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