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Properties

Label 2-234-117.59-c1-0-0
Degree $2$
Conductor $234$
Sign $-0.944 - 0.327i$
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  + (−0.707 + 0.707i)2-s + (−0.769 + 1.55i)3-s − 1.00i·4-s + (0.891 + 3.32i)5-s + (−0.552 − 1.64i)6-s + (−0.00737 − 0.0275i)7-s + (0.707 + 0.707i)8-s + (−1.81 − 2.38i)9-s + (−2.98 − 1.72i)10-s + (3.07 + 3.07i)11-s + (1.55 + 0.769i)12-s + (−3.57 + 0.450i)13-s + (0.0246 + 0.0142i)14-s + (−5.84 − 1.17i)15-s − 1.00·16-s + (−3.35 − 5.81i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.444 + 0.895i)3-s − 0.500i·4-s + (0.398 + 1.48i)5-s + (−0.225 − 0.670i)6-s + (−0.00278 − 0.0104i)7-s + (0.250 + 0.250i)8-s + (−0.604 − 0.796i)9-s + (−0.942 − 0.544i)10-s + (0.927 + 0.927i)11-s + (0.447 + 0.222i)12-s + (−0.992 + 0.124i)13-s + (0.00659 + 0.00380i)14-s + (−1.50 − 0.304i)15-s − 0.250·16-s + (−0.813 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.944 - 0.327i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.944 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128321 + 0.761484i\)
\(L(\frac12)\) \(\approx\) \(0.128321 + 0.761484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.769 - 1.55i)T \)
13 \( 1 + (3.57 - 0.450i)T \)
good5 \( 1 + (-0.891 - 3.32i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.00737 + 0.0275i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.07 - 3.07i)T + 11iT^{2} \)
17 \( 1 + (3.35 + 5.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.521 - 1.94i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.264 - 0.457i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.30iT - 29T^{2} \)
31 \( 1 + (-1.39 + 0.373i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.09 + 7.83i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.85 - 1.30i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-7.58 - 4.38i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.207 - 0.775i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 - 5.06iT - 53T^{2} \)
59 \( 1 + (-10.1 - 10.1i)T + 59iT^{2} \)
61 \( 1 + (-0.0107 + 0.0186i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.244 - 0.912i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-9.32 - 2.49i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.53 + 6.53i)T - 73iT^{2} \)
79 \( 1 + (-1.40 - 2.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (14.1 + 3.80i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (10.8 - 2.91i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.455 - 0.122i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.30431300563974761009158597064, −11.29397938312813312059487210520, −10.54684377839824262815305097580, −9.674863617470727180991934009624, −9.125573696263594631698577688229, −7.21303724457907562898291219979, −6.79016155922344157290162281548, −5.54455956857909121847665580670, −4.26456334785633987991736746416, −2.60011384593120790096551254872, 0.795538554312712897698358499787, 2.15085080540291143040680348232, 4.29754962493616057570931050151, 5.60716381521919923859443698681, 6.66203478799470768071132342011, 8.136161452766893252032551004551, 8.690860280781589790283322545114, 9.701577180426579228866803189027, 10.98306748462270250587111113665, 11.87175195174072428658806112950

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