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Properties

Label 2-234-117.110-c1-0-11
Degree $2$
Conductor $234$
Sign $0.0914 + 0.995i$
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 + (1.71 + 0.208i)3-s + (−0.866 − 0.499i)4-s + (0.406 − 1.51i)5-s + (0.646 − 1.60i)6-s + (−3.31 − 3.31i)7-s + (−0.707 + 0.707i)8-s + (2.91 + 0.718i)9-s + (−1.36 − 0.785i)10-s + (−1.47 − 0.395i)11-s + (−1.38 − 1.04i)12-s + (3.50 + 0.853i)13-s + (−4.05 + 2.34i)14-s + (1.01 − 2.52i)15-s + (0.500 + 0.866i)16-s + (2.99 + 5.18i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.992 + 0.120i)3-s + (−0.433 − 0.249i)4-s + (0.181 − 0.678i)5-s + (0.264 − 0.655i)6-s + (−1.25 − 1.25i)7-s + (−0.249 + 0.249i)8-s + (0.970 + 0.239i)9-s + (−0.430 − 0.248i)10-s + (−0.445 − 0.119i)11-s + (−0.399 − 0.300i)12-s + (0.971 + 0.236i)13-s + (−1.08 + 0.626i)14-s + (0.262 − 0.651i)15-s + (0.125 + 0.216i)16-s + (0.725 + 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0914 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0914 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.0914 + 0.995i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.0914 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21566 - 1.10911i\)
\(L(\frac12)\) \(\approx\) \(1.21566 - 1.10911i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.208i)T \)
13 \( 1 + (-3.50 - 0.853i)T \)
good5 \( 1 + (-0.406 + 1.51i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (3.31 + 3.31i)T + 7iT^{2} \)
11 \( 1 + (1.47 + 0.395i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.99 - 5.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.70 - 1.52i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 4.97T + 23T^{2} \)
29 \( 1 + (5.94 - 3.43i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.01 - 0.271i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (6.94 - 1.86i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.66 + 3.66i)T + 41iT^{2} \)
43 \( 1 + 12.2iT - 43T^{2} \)
47 \( 1 + (-1.64 - 6.14i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 2.78iT - 53T^{2} \)
59 \( 1 + (-1.54 - 5.76i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 4.37T + 61T^{2} \)
67 \( 1 + (-3.96 + 3.96i)T - 67iT^{2} \)
71 \( 1 + (0.0745 - 0.278i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.964 - 0.964i)T + 73iT^{2} \)
79 \( 1 + (0.0673 - 0.116i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.2 - 3.00i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.46 + 5.45i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.71 - 4.71i)T - 97iT^{2} \)
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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

−12.27936144535218744328192941694, −10.62806658488448450981329696087, −10.09156833217167157529909038259, −9.204800569986487588175611559687, −8.227905205967869010422969489783, −7.06186618148774000583044551447, −5.56314352078114110706706148966, −3.88858975795369111121624779843, −3.42287270145729047395033016287, −1.41889348162327999401376958270, 2.70909476653849312932305038513, 3.46056925315022265597811986672, 5.39948396619920582191676626873, 6.41485072025295394748270518274, 7.37800616515877409192310973188, 8.425484161705222946086331772544, 9.462106898541380580068101404513, 9.957878463539164623777835711625, 11.68509967343838973134487855674, 12.70668631654421638202655594730

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