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Properties

Label 2-234-117.11-c1-0-5
Degree $2$
Conductor $234$
Sign $0.722 + 0.690i$
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 + (0.0694 − 1.73i)3-s − 1.00i·4-s + (0.339 + 0.0908i)5-s + (1.17 + 1.27i)6-s + (2.97 + 0.797i)7-s + (0.707 + 0.707i)8-s + (−2.99 − 0.240i)9-s + (−0.304 + 0.175i)10-s + (−2.35 − 2.35i)11-s + (−1.73 − 0.0694i)12-s + (2.60 − 2.49i)13-s + (−2.66 + 1.54i)14-s + (0.180 − 0.580i)15-s − 1.00·16-s + (3.87 − 6.71i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.0401 − 0.999i)3-s − 0.500i·4-s + (0.151 + 0.0406i)5-s + (0.479 + 0.519i)6-s + (1.12 + 0.301i)7-s + (0.250 + 0.250i)8-s + (−0.996 − 0.0801i)9-s + (−0.0961 + 0.0555i)10-s + (−0.709 − 0.709i)11-s + (−0.499 − 0.0200i)12-s + (0.723 − 0.690i)13-s + (−0.713 + 0.411i)14-s + (0.0466 − 0.149i)15-s − 0.250·16-s + (0.940 − 1.62i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.722 + 0.690i$
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: \((2,\ 234,\ (\ :1/2),\ 0.722 + 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.981433 - 0.393538i\)
\(L(\frac12)\) \(\approx\) \(0.981433 - 0.393538i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0694 + 1.73i)T \)
13 \( 1 + (-2.60 + 2.49i)T \)
good5 \( 1 + (-0.339 - 0.0908i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.97 - 0.797i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.35 + 2.35i)T + 11iT^{2} \)
17 \( 1 + (-3.87 + 6.71i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.67 + 0.984i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.34 - 5.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.68iT - 29T^{2} \)
31 \( 1 + (-0.293 + 1.09i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-8.90 - 2.38i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.678 + 2.53i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (3.68 - 2.12i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.17 - 1.11i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + (-7.75 - 7.75i)T + 59iT^{2} \)
61 \( 1 + (-5.01 - 8.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.3 - 3.56i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.343 + 1.28i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (2.19 - 2.19i)T - 73iT^{2} \)
79 \( 1 + (6.22 - 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.05 + 3.92i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-1.43 + 5.37i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.218 - 0.816i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.74949749556696619988535725520, −11.37421569241112146358106105944, −10.02924676429996962181384901750, −8.797401747953847486025448554195, −7.894708346886868507407386716618, −7.43179512899532234650275246483, −5.84885788169194129072004235177, −5.29919743853389043924522807686, −2.88979734604609203307264607820, −1.17567566163927208183146700977, 1.91794542094610249724133875585, 3.72040521366786854538156840335, 4.68263123457604629382979297304, 5.98956188273790638995093170146, 7.86592002634925933768310759044, 8.349143827319299241181797441990, 9.696440507944400960851338306644, 10.27522809827002439799011942449, 11.17243184754610966682383352164, 11.90753967382359902090061691030

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