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Properties

Label 2-234-117.59-c1-0-8
Degree $2$
Conductor $234$
Sign $0.856 + 0.516i$
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 + (1.71 − 0.208i)3-s − 1.00i·4-s + (−0.313 − 1.16i)5-s + (−1.06 + 1.36i)6-s + (−1.16 − 4.35i)7-s + (0.707 + 0.707i)8-s + (2.91 − 0.715i)9-s + (1.04 + 0.604i)10-s + (0.617 + 0.617i)11-s + (−0.208 − 1.71i)12-s + (−1.15 − 3.41i)13-s + (3.90 + 2.25i)14-s + (−0.781 − 1.94i)15-s − 1.00·16-s + (1.49 + 2.58i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.992 − 0.120i)3-s − 0.500i·4-s + (−0.140 − 0.522i)5-s + (−0.436 + 0.556i)6-s + (−0.440 − 1.64i)7-s + (0.250 + 0.250i)8-s + (0.971 − 0.238i)9-s + (0.331 + 0.191i)10-s + (0.186 + 0.186i)11-s + (−0.0600 − 0.496i)12-s + (−0.319 − 0.947i)13-s + (1.04 + 0.601i)14-s + (−0.201 − 0.501i)15-s − 0.250·16-s + (0.362 + 0.627i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.856 + 0.516i$
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: \((2,\ 234,\ (\ :1/2),\ 0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19604 - 0.332475i\)
\(L(\frac12)\) \(\approx\) \(1.19604 - 0.332475i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.208i)T \)
13 \( 1 + (1.15 + 3.41i)T \)
good5 \( 1 + (0.313 + 1.16i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.16 + 4.35i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.617 - 0.617i)T + 11iT^{2} \)
17 \( 1 + (-1.49 - 2.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.92 - 7.17i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.0774 + 0.134i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.39iT - 29T^{2} \)
31 \( 1 + (-4.22 + 1.13i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.18 + 4.40i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.46 - 1.46i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.01 - 1.16i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.69 - 6.31i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + 0.732iT - 53T^{2} \)
59 \( 1 + (10.1 + 10.1i)T + 59iT^{2} \)
61 \( 1 + (1.64 - 2.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.37 - 5.14i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (14.6 + 3.93i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.594 + 0.594i)T - 73iT^{2} \)
79 \( 1 + (-1.78 - 3.09i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.44 + 0.654i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-5.91 + 1.58i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-10.7 + 2.88i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.53994739410256633066096289592, −10.51765214217612687691622794634, −10.18406453884093406156433136333, −9.037846456845746074568351396763, −7.972675586281123793526402001166, −7.46675049138682367715152355119, −6.28659834953122782469788801076, −4.53959682072251135796270153116, −3.43651169819542145907671473995, −1.23498551683818153093438382588, 2.33765028676359115959961147396, 3.02050679193811552519052151666, 4.62351729522225911608566237766, 6.41086524149821699014791172340, 7.47671068393247993469057180272, 8.801088805626011395856915637385, 9.139573227441576439439250786895, 10.07711166915891885150854198746, 11.40394024201952440235026449232, 12.07261213691486494490424132674

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