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Properties

Label 2-234-117.22-c1-0-11
Degree $2$
Conductor $234$
Sign $-0.329 + 0.944i$
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 − 1.73i·3-s + 4-s + (1.5 − 2.59i)5-s + 1.73i·6-s + (0.5 − 0.866i)7-s − 8-s − 2.99·9-s + (−1.5 + 2.59i)10-s − 1.73i·12-s + (1 + 3.46i)13-s + (−0.5 + 0.866i)14-s + (−4.5 − 2.59i)15-s + 16-s + (−1.5 − 2.59i)17-s + 2.99·18-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.999i·3-s + 0.5·4-s + (0.670 − 1.16i)5-s + 0.707i·6-s + (0.188 − 0.327i)7-s − 0.353·8-s − 0.999·9-s + (−0.474 + 0.821i)10-s − 0.499i·12-s + (0.277 + 0.960i)13-s + (−0.133 + 0.231i)14-s + (−1.16 − 0.670i)15-s + 0.250·16-s + (−0.363 − 0.630i)17-s + 0.707·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.329 + 0.944i$
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: \((2,\ 234,\ (\ :1/2),\ -0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548745 - 0.773092i\)
\(L(\frac12)\) \(\approx\) \(0.548745 - 0.773092i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
13 \( 1 + (-1 - 3.46i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.5 - 12.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.02776659183464191278861516197, −10.96907613427425333232802989245, −9.719519435748406270377439762616, −8.791046144755337286342646609256, −8.147201587092163436628687804771, −6.88796779699228937821140960342, −6.01183268293505075064948135086, −4.60937731851655237078049269949, −2.31399535242109755378314829303, −1.03069248005832328975211427130, 2.41514060346324109502670651818, 3.61633223728820169754740593841, 5.47432360237672440155261835779, 6.28518426902667916156014576709, 7.67226700054107553368356085374, 8.764545666429007937560486500874, 9.744585950465248339773846043718, 10.48812332318676872274100453231, 11.01558892482978496562343132860, 12.12212029561317654163808532450

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