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Properties

Label 2-234-117.11-c1-0-2
Degree $2$
Conductor $234$
Sign $0.0797 - 0.996i$
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 + (1.09 + 1.34i)3-s − 1.00i·4-s + (0.994 + 0.266i)5-s + (−1.72 − 0.172i)6-s + (0.339 + 0.0908i)7-s + (0.707 + 0.707i)8-s + (−0.593 + 2.94i)9-s + (−0.891 + 0.514i)10-s + (2.56 + 2.56i)11-s + (1.34 − 1.09i)12-s + (0.0241 − 3.60i)13-s + (−0.304 + 0.175i)14-s + (0.733 + 1.62i)15-s − 1.00·16-s + (−1.67 + 2.89i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.633 + 0.773i)3-s − 0.500i·4-s + (0.444 + 0.119i)5-s + (−0.703 − 0.0702i)6-s + (0.128 + 0.0343i)7-s + (0.250 + 0.250i)8-s + (−0.197 + 0.980i)9-s + (−0.281 + 0.162i)10-s + (0.774 + 0.774i)11-s + (0.386 − 0.316i)12-s + (0.00670 − 0.999i)13-s + (−0.0812 + 0.0469i)14-s + (0.189 + 0.419i)15-s − 0.250·16-s + (−0.405 + 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0797 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0797 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.0797 - 0.996i$
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: \((2,\ 234,\ (\ :1/2),\ 0.0797 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.925943 + 0.854785i\)
\(L(\frac12)\) \(\approx\) \(0.925943 + 0.854785i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 - 1.34i)T \)
13 \( 1 + (-0.0241 + 3.60i)T \)
good5 \( 1 + (-0.994 - 0.266i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.339 - 0.0908i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.56 - 2.56i)T + 11iT^{2} \)
17 \( 1 + (1.67 - 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.969 - 0.259i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.735 + 1.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.82iT - 29T^{2} \)
31 \( 1 + (-1.48 + 5.54i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-4.96 - 1.33i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.41 + 9.02i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.37 + 1.36i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.18 + 2.19i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 2.67iT - 53T^{2} \)
59 \( 1 + (3.42 + 3.42i)T + 59iT^{2} \)
61 \( 1 + (2.86 + 4.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-14.5 + 3.90i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.53 + 5.74i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.47 + 2.47i)T - 73iT^{2} \)
79 \( 1 + (1.76 - 3.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.23 + 8.34i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (2.68 - 10.0i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.82 - 14.2i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.48929785841482736790763546123, −11.02448551524699784409956814022, −10.22444611187654467036514583267, −9.475514618008991854409688840423, −8.572130313288803638289759967352, −7.65668289946992788389127027698, −6.36851845575163992153159333233, −5.16559157811899310211207644119, −3.89841582326666725855230575125, −2.15228824800201216837612273747, 1.36566893642900256384397611381, 2.71942625461287716317767960298, 4.15767248111890587573259118153, 6.09049888882828651099430018853, 7.05615734812990329188744533098, 8.193552908891893628516803377513, 9.108902171129071638703062843696, 9.667725822911483745186025760455, 11.25832471239876758416408259277, 11.78232000109473388721451662265

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