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Properties

Label 2-234-117.103-c1-0-4
Degree $2$
Conductor $234$
Sign $0.876 - 0.480i$
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  + (−0.866 + 0.5i)2-s + (1.66 + 0.480i)3-s + (0.499 − 0.866i)4-s + (−0.515 − 0.297i)5-s + (−1.68 + 0.416i)6-s + (1.45 − 0.838i)7-s + 0.999i·8-s + (2.53 + 1.59i)9-s + 0.594·10-s + (0.416 − 0.240i)11-s + (1.24 − 1.20i)12-s + (3.56 + 0.567i)13-s + (−0.838 + 1.45i)14-s + (−0.714 − 0.742i)15-s + (−0.5 − 0.866i)16-s − 2.09·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.960 + 0.277i)3-s + (0.249 − 0.433i)4-s + (−0.230 − 0.132i)5-s + (−0.686 + 0.169i)6-s + (0.548 − 0.316i)7-s + 0.353i·8-s + (0.846 + 0.532i)9-s + 0.188·10-s + (0.125 − 0.0724i)11-s + (0.360 − 0.346i)12-s + (0.987 + 0.157i)13-s + (−0.224 + 0.388i)14-s + (−0.184 − 0.191i)15-s + (−0.125 − 0.216i)16-s − 0.507·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.876 - 0.480i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.876 - 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26241 + 0.323473i\)
\(L(\frac12)\) \(\approx\) \(1.26241 + 0.323473i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-1.66 - 0.480i)T \)
13 \( 1 + (-3.56 - 0.567i)T \)
good5 \( 1 + (0.515 + 0.297i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.45 + 0.838i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.416 + 0.240i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.09T + 17T^{2} \)
19 \( 1 - 0.480iT - 19T^{2} \)
23 \( 1 + (1.83 - 3.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.993 - 0.573i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.65iT - 37T^{2} \)
41 \( 1 + (8.58 + 4.95i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.45 + 5.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.40 + 3.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.08T + 53T^{2} \)
59 \( 1 + (8.13 + 4.69i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.90 + 6.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.4 + 7.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.51iT - 71T^{2} \)
73 \( 1 - 5.91iT - 73T^{2} \)
79 \( 1 + (-1.02 - 1.78i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.57 + 5.53i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.48iT - 89T^{2} \)
97 \( 1 + (8.41 - 4.85i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.15947108538098577119613966911, −11.00506977580328858167945082790, −10.20278776031439760101456306338, −9.093928886973036730687694006875, −8.372329628705617652514682524934, −7.60774482108058868438316370085, −6.41093213316357803959301581376, −4.80332836639551488903916927207, −3.57659992932741169850510216918, −1.74362787917034454929791003531, 1.65648706682890724467306018636, 3.05670726606791139591779686467, 4.33102106595955063222825381395, 6.25877615854497565126294005558, 7.46213402289725356525230600673, 8.333648825349041702359355376856, 8.982266550062208200349161800730, 10.05874815959061000798262424085, 11.12968366659057809547253226598, 11.98376744317546736274775359676

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