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Properties

Label 2-234-117.20-c1-0-2
Degree $2$
Conductor $234$
Sign $-0.887 - 0.461i$
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 + (−1.09 + 1.34i)3-s + (0.866 + 0.499i)4-s + (−3.60 − 0.964i)5-s + (−1.40 + 1.01i)6-s + (−2.35 + 2.35i)7-s + (0.707 + 0.707i)8-s + (−0.599 − 2.93i)9-s + (−3.22 − 1.86i)10-s + (−1.64 + 6.12i)11-s + (−1.61 + 0.613i)12-s + (3.47 + 0.961i)13-s + (−2.88 + 1.66i)14-s + (5.24 − 3.77i)15-s + (0.500 + 0.866i)16-s + (−1.38 − 2.39i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.632 + 0.774i)3-s + (0.433 + 0.249i)4-s + (−1.61 − 0.431i)5-s + (−0.573 + 0.413i)6-s + (−0.889 + 0.889i)7-s + (0.249 + 0.249i)8-s + (−0.199 − 0.979i)9-s + (−1.02 − 0.589i)10-s + (−0.494 + 1.84i)11-s + (−0.467 + 0.177i)12-s + (0.963 + 0.266i)13-s + (−0.769 + 0.444i)14-s + (1.35 − 0.974i)15-s + (0.125 + 0.216i)16-s + (−0.335 − 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.887 - 0.461i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.887 - 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.176853 + 0.723684i\)
\(L(\frac12)\) \(\approx\) \(0.176853 + 0.723684i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 - 1.34i)T \)
13 \( 1 + (-3.47 - 0.961i)T \)
good5 \( 1 + (3.60 + 0.964i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.35 - 2.35i)T - 7iT^{2} \)
11 \( 1 + (1.64 - 6.12i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.38 + 2.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.703 + 2.62i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.183T + 23T^{2} \)
29 \( 1 + (0.821 - 0.474i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.573 - 2.14i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.96 - 7.33i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.66 + 1.66i)T - 41iT^{2} \)
43 \( 1 - 3.66iT - 43T^{2} \)
47 \( 1 + (5.15 - 1.38i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 3.43iT - 53T^{2} \)
59 \( 1 + (-4.52 + 1.21i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 8.27T + 61T^{2} \)
67 \( 1 + (-5.28 - 5.28i)T + 67iT^{2} \)
71 \( 1 + (11.7 + 3.16i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.16 + 2.16i)T - 73iT^{2} \)
79 \( 1 + (-6.31 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.74 - 6.52i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.643 - 0.172i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.14 - 4.14i)T + 97iT^{2} \)
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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

−12.36204684649530552273480236997, −11.84158002652454264297502303987, −10.99680564431166567325523661301, −9.689501799359363022659425579999, −8.740176208959732132701393943982, −7.39669052701327972130288355771, −6.40561901388881462227709564055, −4.99439347585902213542698995374, −4.33527762776358631473509162944, −3.13122471433430396792119755470, 0.52299511128173477469339652469, 3.23758659845593694285486361192, 3.96413372810773840295477826826, 5.74884974806545445032461017376, 6.57768205897799290737660582450, 7.61701055721284116290931533859, 8.380027466422139227614214866100, 10.55805115679432922613713847820, 11.01071227939458687939983495725, 11.69440117944489197870961317423

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