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Properties

Label 2-234-117.50-c1-0-8
Degree $2$
Conductor $234$
Sign $0.744 + 0.667i$
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  + (0.258 + 0.965i)2-s + (−1.53 + 0.810i)3-s + (−0.866 + 0.499i)4-s + (−0.970 − 3.62i)5-s + (−1.17 − 1.26i)6-s + (2.29 − 2.29i)7-s + (−0.707 − 0.707i)8-s + (1.68 − 2.48i)9-s + (3.24 − 1.87i)10-s + (−2.11 + 0.566i)11-s + (0.920 − 1.46i)12-s + (−3.26 + 1.53i)13-s + (2.81 + 1.62i)14-s + (4.42 + 4.75i)15-s + (0.500 − 0.866i)16-s + (3.50 − 6.07i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.883 + 0.467i)3-s + (−0.433 + 0.249i)4-s + (−0.433 − 1.61i)5-s + (−0.481 − 0.517i)6-s + (0.868 − 0.868i)7-s + (−0.249 − 0.249i)8-s + (0.562 − 0.827i)9-s + (1.02 − 0.592i)10-s + (−0.637 + 0.170i)11-s + (0.265 − 0.423i)12-s + (−0.905 + 0.424i)13-s + (0.751 + 0.434i)14-s + (1.14 + 1.22i)15-s + (0.125 − 0.216i)16-s + (0.850 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.744 + 0.667i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.744 + 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.777370 - 0.297235i\)
\(L(\frac12)\) \(\approx\) \(0.777370 - 0.297235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (1.53 - 0.810i)T \)
13 \( 1 + (3.26 - 1.53i)T \)
good5 \( 1 + (0.970 + 3.62i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.29 + 2.29i)T - 7iT^{2} \)
11 \( 1 + (2.11 - 0.566i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.50 + 6.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.70 + 1.26i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.831T + 23T^{2} \)
29 \( 1 + (8.02 + 4.63i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.527 + 0.141i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.90 - 0.509i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.24 - 1.24i)T - 41iT^{2} \)
43 \( 1 - 0.0581iT - 43T^{2} \)
47 \( 1 + (-1.56 + 5.83i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 - 1.62iT - 53T^{2} \)
59 \( 1 + (0.755 - 2.81i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 3.57T + 61T^{2} \)
67 \( 1 + (-9.19 - 9.19i)T + 67iT^{2} \)
71 \( 1 + (-2.31 - 8.65i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.38 + 3.38i)T - 73iT^{2} \)
79 \( 1 + (5.86 + 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.0 - 3.75i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (3.50 - 13.0i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-6.97 - 6.97i)T + 97iT^{2} \)
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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

−11.95779340420232499485124719877, −11.44733082099355310536843325734, −9.912782656054874433429330366167, −9.222755105277459584260203440069, −7.80798372938560600837692037253, −7.26248647746236258255433848750, −5.30811500796894446661582709341, −5.02209291776911377597755338632, −4.06883363584553074411071308856, −0.74559027121493917260602139422, 2.05789489265320849202748438986, 3.37808122360749591924476698695, 5.15184951345527431312151920047, 5.94876355439971739593482235685, 7.37324409416045714094078156030, 8.044173267666069847945870484700, 9.891155937347618547172116800356, 10.74330238637134024113342888274, 11.27418521074512839893744123626, 12.11512764616403679113706288537

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