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Properties

Label 2-234-117.11-c1-0-1
Degree $2$
Conductor $234$
Sign $-0.122 - 0.992i$
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 + (−1.04 + 1.38i)3-s − 1.00i·4-s + (1.51 + 0.406i)5-s + (−0.243 − 1.71i)6-s + (4.52 + 1.21i)7-s + (0.707 + 0.707i)8-s + (−0.834 − 2.88i)9-s + (−1.36 + 0.785i)10-s + (−1.08 − 1.08i)11-s + (1.38 + 1.04i)12-s + (−1.01 + 3.46i)13-s + (−4.05 + 2.34i)14-s + (−2.14 + 1.67i)15-s − 1.00·16-s + (−2.99 + 5.18i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.600 + 0.799i)3-s − 0.500i·4-s + (0.678 + 0.181i)5-s + (−0.0993 − 0.700i)6-s + (1.71 + 0.458i)7-s + (0.250 + 0.250i)8-s + (−0.278 − 0.960i)9-s + (−0.430 + 0.248i)10-s + (−0.326 − 0.326i)11-s + (0.399 + 0.300i)12-s + (−0.280 + 0.959i)13-s + (−1.08 + 0.626i)14-s + (−0.552 + 0.433i)15-s − 0.250·16-s + (−0.725 + 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.122 - 0.992i$
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: \((2,\ 234,\ (\ :1/2),\ -0.122 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.651392 + 0.736894i\)
\(L(\frac12)\) \(\approx\) \(0.651392 + 0.736894i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.04 - 1.38i)T \)
13 \( 1 + (1.01 - 3.46i)T \)
good5 \( 1 + (-1.51 - 0.406i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-4.52 - 1.21i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.08 + 1.08i)T + 11iT^{2} \)
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 \( 1 + (2.48 - 4.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.86iT - 29T^{2} \)
31 \( 1 + (0.271 - 1.01i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (6.94 + 1.86i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.34 - 5.00i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-10.6 + 6.12i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.14 + 1.64i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 2.78iT - 53T^{2} \)
59 \( 1 + (4.22 + 4.22i)T + 59iT^{2} \)
61 \( 1 + (2.18 + 3.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.41 - 1.45i)T + (58.0 - 33.5i)T^{2} \)
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 \( 1 + (3.00 + 11.2i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-1.46 + 5.45i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.72 - 6.44i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.97898780026842100518530102154, −11.30866747513275190443572961755, −10.49889250289969143507377803743, −9.482942756561868129415388615955, −8.665519260380290080244542803938, −7.54921427566294499930724601645, −6.08440909848499261128061119431, −5.37815506349575705795303924486, −4.29237360266272092147884442672, −1.91026515153173549653959407435, 1.16800219606034841704185782218, 2.41708590897414193419382295711, 4.76003520928271690539308840980, 5.55910925573866853985781879658, 7.29627203475762711189539078400, 7.76897684586577307551295440983, 8.959853539289331088610048337530, 10.29745239419458163420899745180, 10.95541718596267555237442021920, 11.84098939060121549316915878155

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