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Properties

Label 2-234-117.110-c1-0-12
Degree $2$
Conductor $234$
Sign $-0.618 + 0.785i$
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 + (0.612 − 1.62i)3-s + (−0.866 − 0.499i)4-s + (0.266 − 0.994i)5-s + (−1.40 − 1.01i)6-s + (−0.248 − 0.248i)7-s + (−0.707 + 0.707i)8-s + (−2.25 − 1.98i)9-s + (−0.891 − 0.514i)10-s + (3.50 + 0.940i)11-s + (−1.34 + 1.09i)12-s + (−3.13 − 1.78i)13-s + (−0.304 + 0.175i)14-s + (−1.44 − 1.04i)15-s + (0.500 + 0.866i)16-s + (1.67 + 2.89i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.353 − 0.935i)3-s + (−0.433 − 0.249i)4-s + (0.119 − 0.444i)5-s + (−0.574 − 0.412i)6-s + (−0.0938 − 0.0938i)7-s + (−0.249 + 0.249i)8-s + (−0.750 − 0.661i)9-s + (−0.281 − 0.162i)10-s + (1.05 + 0.283i)11-s + (−0.386 + 0.316i)12-s + (−0.869 − 0.494i)13-s + (−0.0812 + 0.0469i)14-s + (−0.373 − 0.268i)15-s + (0.125 + 0.216i)16-s + (0.405 + 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.618 + 0.785i$
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: \((2,\ 234,\ (\ :1/2),\ -0.618 + 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615614 - 1.26874i\)
\(L(\frac12)\) \(\approx\) \(0.615614 - 1.26874i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.612 + 1.62i)T \)
13 \( 1 + (3.13 + 1.78i)T \)
good5 \( 1 + (-0.266 + 0.994i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.248 + 0.248i)T + 7iT^{2} \)
11 \( 1 + (-3.50 - 0.940i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.67 - 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.969 + 0.259i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 1.47T + 23T^{2} \)
29 \( 1 + (-5.04 + 2.91i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.54 + 1.48i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-4.96 + 1.33i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-6.60 - 6.60i)T + 41iT^{2} \)
43 \( 1 + 2.73iT - 43T^{2} \)
47 \( 1 + (-2.19 - 8.18i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 2.67iT - 53T^{2} \)
59 \( 1 + (-1.25 - 4.68i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 5.72T + 61T^{2} \)
67 \( 1 + (10.6 - 10.6i)T - 67iT^{2} \)
71 \( 1 + (-1.53 + 5.74i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.47 - 2.47i)T + 73iT^{2} \)
79 \( 1 + (1.76 - 3.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.34 - 2.23i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.68 - 10.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (10.4 - 10.4i)T - 97iT^{2} \)
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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

−12.12305852042895965753119217600, −11.10156288497488388711053538180, −9.802706574562793886441787499687, −8.998515966581591923205205234059, −7.953836608665253661897155144583, −6.79938390694241194317041118794, −5.61014908642024320684189845886, −4.13923567803232618211366536862, −2.69300163201121659080382543687, −1.21490433335621926053983864463, 2.85413592133069631733765040977, 4.12784840101191658058019795142, 5.19005063880002375009446393014, 6.44642763776826629097581905689, 7.46964233906278222222676914360, 8.803430612881706711051999142942, 9.403233833642067133073081277155, 10.43361577886488166408036408488, 11.51260092935853720906174570595, 12.53923988988495121527792832700

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