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Properties

Label 2-234-117.110-c1-0-13
Degree $2$
Conductor $234$
Sign $-0.943 + 0.330i$
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 + (−1.22 − 1.22i)3-s + (−0.866 − 0.499i)4-s + (1.02 − 3.83i)5-s + (−1.50 + 0.862i)6-s + (1.13 + 1.13i)7-s + (−0.707 + 0.707i)8-s + (−0.0126 + 2.99i)9-s + (−3.43 − 1.98i)10-s + (−4.79 − 1.28i)11-s + (0.444 + 1.67i)12-s + (3.50 + 0.826i)13-s + (1.39 − 0.803i)14-s + (−5.95 + 3.42i)15-s + (0.500 + 0.866i)16-s + (−0.584 − 1.01i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.705 − 0.708i)3-s + (−0.433 − 0.249i)4-s + (0.459 − 1.71i)5-s + (−0.613 + 0.352i)6-s + (0.429 + 0.429i)7-s + (−0.249 + 0.249i)8-s + (−0.00423 + 0.999i)9-s + (−1.08 − 0.627i)10-s + (−1.44 − 0.387i)11-s + (0.128 + 0.483i)12-s + (0.973 + 0.229i)13-s + (0.371 − 0.214i)14-s + (−1.53 + 0.883i)15-s + (0.125 + 0.216i)16-s + (−0.141 − 0.245i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.943 + 0.330i$
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: \((2,\ 234,\ (\ :1/2),\ -0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171636 - 1.00970i\)
\(L(\frac12)\) \(\approx\) \(0.171636 - 1.00970i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 + 1.22i)T \)
13 \( 1 + (-3.50 - 0.826i)T \)
good5 \( 1 + (-1.02 + 3.83i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.13 - 1.13i)T + 7iT^{2} \)
11 \( 1 + (4.79 + 1.28i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.584 + 1.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.16 + 1.11i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 3.26T + 23T^{2} \)
29 \( 1 + (-6.50 + 3.75i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.25 - 1.94i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.28 - 1.14i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.96 - 3.96i)T + 41iT^{2} \)
43 \( 1 - 0.889iT - 43T^{2} \)
47 \( 1 + (-0.653 - 2.43i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 6.60iT - 53T^{2} \)
59 \( 1 + (2.12 + 7.94i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 + (-2.33 + 2.33i)T - 67iT^{2} \)
71 \( 1 + (0.942 - 3.51i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.33 + 3.33i)T + 73iT^{2} \)
79 \( 1 + (1.16 - 2.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.5 - 2.82i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-1.78 - 6.65i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.52 - 4.52i)T - 97iT^{2} \)
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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

−11.86436049430502714073475129430, −11.04039771829205781906936585022, −9.999543171211348060917981239790, −8.538636750451539453689116783615, −8.274637000525684924458017045023, −6.29191461329450677152164335018, −5.25864331749796268954659293958, −4.67726967370277872977054707026, −2.30219047117469223151515463721, −0.906112113735969767480484395294, 2.92828853269556760051384227960, 4.29992620083932690138460538484, 5.58628938669139488601429362044, 6.44628196813613205479982037323, 7.32411883871353106782419090098, 8.583885237002034899044042260597, 10.25109982947113481202887633705, 10.45309831808155778528269567741, 11.29036485697885812888988789850, 12.72278892416242944694038509669

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