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Properties

Label 2-234-117.11-c1-0-3
Degree $2$
Conductor $234$
Sign $0.983 - 0.179i$
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 + (−0.870 + 1.49i)3-s − 1.00i·4-s + (1.35 + 0.361i)5-s + (0.443 + 1.67i)6-s + (0.977 + 0.261i)7-s + (−0.707 − 0.707i)8-s + (−1.48 − 2.60i)9-s + (1.21 − 0.699i)10-s + (4.11 + 4.11i)11-s + (1.49 + 0.870i)12-s + (3.22 + 1.61i)13-s + (0.876 − 0.505i)14-s + (−1.71 + 1.70i)15-s − 1.00·16-s + (1.67 − 2.89i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.502 + 0.864i)3-s − 0.500i·4-s + (0.603 + 0.161i)5-s + (0.181 + 0.683i)6-s + (0.369 + 0.0989i)7-s + (−0.250 − 0.250i)8-s + (−0.495 − 0.868i)9-s + (0.382 − 0.221i)10-s + (1.23 + 1.23i)11-s + (0.432 + 0.251i)12-s + (0.894 + 0.446i)13-s + (0.234 − 0.135i)14-s + (−0.443 + 0.440i)15-s − 0.250·16-s + (0.405 − 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.983 - 0.179i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.983 - 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54859 + 0.140352i\)
\(L(\frac12)\) \(\approx\) \(1.54859 + 0.140352i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.870 - 1.49i)T \)
13 \( 1 + (-3.22 - 1.61i)T \)
good5 \( 1 + (-1.35 - 0.361i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.977 - 0.261i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-4.11 - 4.11i)T + 11iT^{2} \)
17 \( 1 + (-1.67 + 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.07 - 1.89i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.290 + 0.502i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.03iT - 29T^{2} \)
31 \( 1 + (-2.28 + 8.52i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (7.82 + 2.09i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.423 + 1.58i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.25 - 4.19i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.45 - 0.388i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 2.77iT - 53T^{2} \)
59 \( 1 + (-8.47 - 8.47i)T + 59iT^{2} \)
61 \( 1 + (1.41 + 2.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.09 + 0.293i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.71 - 10.1i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.788 + 0.788i)T - 73iT^{2} \)
79 \( 1 + (0.827 - 1.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.21 + 15.7i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.783 - 2.92i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.18 + 11.8i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.91334742720064152126523637032, −11.41663325040074021492521604780, −10.24830466052153491607211008681, −9.671898041223834010684976674324, −8.654801087883587455183825068418, −6.71102312323835956747838284233, −5.92353229949352875773131876923, −4.62559975197487510471457191188, −3.83871545125913533021707248339, −1.92859565872646158159541839642, 1.53547011328581311839608422135, 3.55140502850772401365532536949, 5.16762334103507967552124860920, 6.17920425991879360115924543147, 6.67966103765312048767918541948, 8.252257869643683840679284076673, 8.732520750329910691634731244379, 10.55296456368632994244666334610, 11.34922579582421332551575976199, 12.31545427754944169702169128919

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