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Properties

Label 2-234-117.103-c1-0-0
Degree $2$
Conductor $234$
Sign $-0.865 + 0.501i$
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  + (−0.866 + 0.5i)2-s + (−1.62 + 0.612i)3-s + (0.499 − 0.866i)4-s + (0.548 + 0.316i)5-s + (1.09 − 1.34i)6-s + (−2.15 + 1.24i)7-s + 0.999i·8-s + (2.24 − 1.98i)9-s − 0.633·10-s + (−4.20 + 2.43i)11-s + (−0.279 + 1.70i)12-s + (−0.541 − 3.56i)13-s + (1.24 − 2.15i)14-s + (−1.08 − 0.176i)15-s + (−0.5 − 0.866i)16-s − 6.27·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.935 + 0.353i)3-s + (0.249 − 0.433i)4-s + (0.245 + 0.141i)5-s + (0.447 − 0.547i)6-s + (−0.814 + 0.469i)7-s + 0.353i·8-s + (0.749 − 0.661i)9-s − 0.200·10-s + (−1.26 + 0.732i)11-s + (−0.0806 + 0.493i)12-s + (−0.150 − 0.988i)13-s + (0.332 − 0.575i)14-s + (−0.279 − 0.0456i)15-s + (−0.125 − 0.216i)16-s − 1.52·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.865 + 0.501i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.865 + 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0164801 - 0.0613559i\)
\(L(\frac12)\) \(\approx\) \(0.0164801 - 0.0613559i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (1.62 - 0.612i)T \)
13 \( 1 + (0.541 + 3.56i)T \)
good5 \( 1 + (-0.548 - 0.316i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.15 - 1.24i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.20 - 2.43i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 6.27T + 17T^{2} \)
19 \( 1 + 4.86iT - 19T^{2} \)
23 \( 1 + (1.77 - 3.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.415 - 0.719i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.73 + 2.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.81iT - 37T^{2} \)
41 \( 1 + (-0.0678 - 0.0391i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.84 - 8.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.30 + 2.48i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.36T + 53T^{2} \)
59 \( 1 + (-7.86 - 4.54i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.28 - 9.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.82 + 3.94i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 - 1.05iT - 73T^{2} \)
79 \( 1 + (1.68 + 2.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (13.1 - 7.60i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.595iT - 89T^{2} \)
97 \( 1 + (-14.7 + 8.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.76352972109650081417696104440, −11.55685884376098180003160271863, −10.58487075639055986094949724080, −9.941389183062139354725683164570, −9.074201843266751410366535498106, −7.66549485121842706128334343858, −6.61782482537019609845335985198, −5.72773206966556483602269880846, −4.67918804309072372810674431570, −2.60166767609160998247309073477, 0.06472313265520329611322090094, 2.12506133563451912212272979153, 4.01154176665103283429830954797, 5.56263424046431513599858521928, 6.61106315231499250004758337136, 7.52224730974668955739434666437, 8.777664799135788031412956359408, 9.912805738789492771777381513699, 10.70339795454305204597829738759, 11.39163348368467541403563190173

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