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Properties

Label 2-234-13.8-c2-0-9
Degree $2$
Conductor $234$
Sign $-0.697 + 0.716i$
Analytic cond. $6.37603$
Root an. cond. $2.52508$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + 2i·4-s + (−1.26 − 1.26i)5-s + (0.732 − 0.732i)7-s + (2 − 2i)8-s + 2.53i·10-s + (1.73 − 1.73i)11-s + (−3.92 − 12.3i)13-s − 1.46·14-s − 4·16-s − 5.32i·17-s + (−14.7 − 14.7i)19-s + (2.53 − 2.53i)20-s − 3.46·22-s − 5.32i·23-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 0.5i·4-s + (−0.253 − 0.253i)5-s + (0.104 − 0.104i)7-s + (0.250 − 0.250i)8-s + 0.253i·10-s + (0.157 − 0.157i)11-s + (−0.302 − 0.953i)13-s − 0.104·14-s − 0.250·16-s − 0.312i·17-s + (−0.775 − 0.775i)19-s + (0.126 − 0.126i)20-s − 0.157·22-s − 0.231i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.697 + 0.716i$
Analytic conductor: \(6.37603\)
Root analytic conductor: \(2.52508\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1),\ -0.697 + 0.716i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.313540 - 0.743287i\)
\(L(\frac12)\) \(\approx\) \(0.313540 - 0.743287i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 \)
13 \( 1 + (3.92 + 12.3i)T \)
good5 \( 1 + (1.26 + 1.26i)T + 25iT^{2} \)
7 \( 1 + (-0.732 + 0.732i)T - 49iT^{2} \)
11 \( 1 + (-1.73 + 1.73i)T - 121iT^{2} \)
17 \( 1 + 5.32iT - 289T^{2} \)
19 \( 1 + (14.7 + 14.7i)T + 361iT^{2} \)
23 \( 1 + 5.32iT - 529T^{2} \)
29 \( 1 + 4.14T + 841T^{2} \)
31 \( 1 + (24.9 + 24.9i)T + 961iT^{2} \)
37 \( 1 + (3.14 - 3.14i)T - 1.36e3iT^{2} \)
41 \( 1 + (44.4 + 44.4i)T + 1.68e3iT^{2} \)
43 \( 1 - 37.1iT - 1.84e3T^{2} \)
47 \( 1 + (-30.8 + 30.8i)T - 2.20e3iT^{2} \)
53 \( 1 + 57.7T + 2.80e3T^{2} \)
59 \( 1 + (-66.6 + 66.6i)T - 3.48e3iT^{2} \)
61 \( 1 - 103.T + 3.72e3T^{2} \)
67 \( 1 + (-46.6 - 46.6i)T + 4.48e3iT^{2} \)
71 \( 1 + (-26.9 - 26.9i)T + 5.04e3iT^{2} \)
73 \( 1 + (-5.67 + 5.67i)T - 5.32e3iT^{2} \)
79 \( 1 - 4.21T + 6.24e3T^{2} \)
83 \( 1 + (-109. - 109. i)T + 6.88e3iT^{2} \)
89 \( 1 + (-19.5 + 19.5i)T - 7.92e3iT^{2} \)
97 \( 1 + (-4.03 - 4.03i)T + 9.40e3iT^{2} \)
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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

−11.47132073247722834006482497870, −10.66074312551139075731873208003, −9.709220883198772866944416309922, −8.660075508999995201086912042234, −7.85706246542587286524730281549, −6.69036102271831901637259441600, −5.17488004732636069661576388779, −3.87624092252189187114772470348, −2.41954779898275227915556637243, −0.50512935395451794411855705683, 1.83247899922811176691556890435, 3.78616070407071865192926234295, 5.17372006142428879547519294613, 6.45322013186438408046051707309, 7.29863088961187822551892446328, 8.380113895688492184540607742354, 9.282630351195077745514968765140, 10.28493619347200501781732463545, 11.24947868703277446581628447592, 12.17155968997390986094408494187

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