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Properties

Label 2-234-117.22-c1-0-0
Degree $2$
Conductor $234$
Sign $-0.524 - 0.851i$
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 + (0.355 − 1.69i)3-s + 4-s + (−2.14 + 3.71i)5-s + (−0.355 + 1.69i)6-s + (−0.751 + 1.30i)7-s − 8-s + (−2.74 − 1.20i)9-s + (2.14 − 3.71i)10-s − 4.24·11-s + (0.355 − 1.69i)12-s + (−3.60 + 0.0419i)13-s + (0.751 − 1.30i)14-s + (5.53 + 4.95i)15-s + 16-s + (0.0242 + 0.0419i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.205 − 0.978i)3-s + 0.5·4-s + (−0.959 + 1.66i)5-s + (−0.145 + 0.692i)6-s + (−0.283 + 0.491i)7-s − 0.353·8-s + (−0.915 − 0.401i)9-s + (0.678 − 1.17i)10-s − 1.27·11-s + (0.102 − 0.489i)12-s + (−0.999 + 0.0116i)13-s + (0.200 − 0.347i)14-s + (1.42 + 1.28i)15-s + 0.250·16-s + (0.00588 + 0.0101i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.524 - 0.851i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.524 - 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174530 + 0.312498i\)
\(L(\frac12)\) \(\approx\) \(0.174530 + 0.312498i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.355 + 1.69i)T \)
13 \( 1 + (3.60 - 0.0419i)T \)
good5 \( 1 + (2.14 - 3.71i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.751 - 1.30i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
17 \( 1 + (-0.0242 - 0.0419i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.87 - 4.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.41 - 5.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.21T + 29T^{2} \)
31 \( 1 + (-3.74 + 6.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.35 - 2.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.68 + 6.39i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.89 - 5.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.67 + 2.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.82T + 53T^{2} \)
59 \( 1 + 1.88T + 59T^{2} \)
61 \( 1 + (0.877 - 1.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.73 - 3.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.645 - 1.11i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 + (-4.24 - 7.35i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.370 + 0.642i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.288 + 0.500i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.59 - 4.49i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.11785414099702228182275891134, −11.64450510524164948299705121870, −10.58705253788001170608028038460, −9.704637346843388628228111507074, −8.136134638401834129840686700193, −7.59413537667196511401541395889, −6.89410507032120786690619282995, −5.71670636010093322481053712890, −3.25107678388294296367734290051, −2.46553784938447862938419620179, 0.33418551454119104893748572633, 3.00233180854061555298963123271, 4.62768563717673725574847339213, 5.13113865046407338699067204641, 7.23053351468390972986131250972, 8.232162130051598721694807537792, 8.857129603668757029466513078222, 9.821675864538037933939737611147, 10.67684149288535506598494600378, 11.74471053367781831228803227597

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