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Properties

Label 2-234-117.16-c1-0-0
Degree $2$
Conductor $234$
Sign $-0.656 - 0.754i$
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  − 2-s + (−1.65 + 0.499i)3-s + 4-s + (−0.102 − 0.178i)5-s + (1.65 − 0.499i)6-s + (0.779 + 1.35i)7-s − 8-s + (2.50 − 1.65i)9-s + (0.102 + 0.178i)10-s − 3.88·11-s + (−1.65 + 0.499i)12-s + (−1.68 + 3.18i)13-s + (−0.779 − 1.35i)14-s + (0.259 + 0.244i)15-s + 16-s + (−1.84 + 3.18i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.957 + 0.288i)3-s + 0.5·4-s + (−0.0460 − 0.0797i)5-s + (0.677 − 0.203i)6-s + (0.294 + 0.510i)7-s − 0.353·8-s + (0.833 − 0.551i)9-s + (0.0325 + 0.0563i)10-s − 1.17·11-s + (−0.478 + 0.144i)12-s + (−0.466 + 0.884i)13-s + (−0.208 − 0.360i)14-s + (0.0670 + 0.0630i)15-s + 0.250·16-s + (−0.446 + 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.656 - 0.754i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.155481 + 0.341645i\)
\(L(\frac12)\) \(\approx\) \(0.155481 + 0.341645i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.65 - 0.499i)T \)
13 \( 1 + (1.68 - 3.18i)T \)
good5 \( 1 + (0.102 + 0.178i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.779 - 1.35i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.88T + 11T^{2} \)
17 \( 1 + (1.84 - 3.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.72 - 6.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.68 - 2.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.87T + 29T^{2} \)
31 \( 1 + (1.50 + 2.60i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.38 - 7.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.74 + 3.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.343 - 0.594i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.48 - 2.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.52T + 53T^{2} \)
59 \( 1 - 6.60T + 59T^{2} \)
61 \( 1 + (2.67 + 4.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.51 + 2.62i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.39 - 2.41i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 + (-0.829 + 1.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.34 - 7.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.25 - 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.28 + 5.69i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.24561555320474539642322571267, −11.45807546802457485702171979579, −10.52072144487744950681091801933, −9.853579694570048787153971631947, −8.642919637732344743653716166107, −7.65714703977333890022311606573, −6.38727168411325115212039639386, −5.47990439525814649598198540237, −4.16261128148640139560973716064, −2.02087366323342749339782934527, 0.40863396697664379965301892169, 2.49055900416094205306143586238, 4.65652881888126073067874324097, 5.68056011189308169850285857486, 7.09889995217992203338884285068, 7.56471904600405084590744646109, 8.893565570615915970882116879759, 10.21805078008616449525888519147, 10.81489885610018845305654695457, 11.49204751960979107037212699677

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