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Properties

Label 2-230-115.68-c1-0-3
Degree $2$
Conductor $230$
Sign $0.999 - 0.00955i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
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 + (0.575 + 0.575i)3-s − 1.00i·4-s + (−0.185 + 2.22i)5-s + 0.814·6-s + (2.09 + 2.09i)7-s + (−0.707 − 0.707i)8-s − 2.33i·9-s + (1.44 + 1.70i)10-s − 1.39i·11-s + (0.575 − 0.575i)12-s + (4.24 + 4.24i)13-s + 2.96·14-s + (−1.38 + 1.17i)15-s − 1.00·16-s + (−4.38 − 4.38i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.332 + 0.332i)3-s − 0.500i·4-s + (−0.0831 + 0.996i)5-s + 0.332·6-s + (0.792 + 0.792i)7-s + (−0.250 − 0.250i)8-s − 0.779i·9-s + (0.456 + 0.539i)10-s − 0.422i·11-s + (0.166 − 0.166i)12-s + (1.17 + 1.17i)13-s + 0.792·14-s + (−0.358 + 0.303i)15-s − 0.250·16-s + (−1.06 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.999 - 0.00955i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.999 - 0.00955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78843 + 0.00854678i\)
\(L(\frac12)\) \(\approx\) \(1.78843 + 0.00854678i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.185 - 2.22i)T \)
23 \( 1 + (-0.664 + 4.74i)T \)
good3 \( 1 + (-0.575 - 0.575i)T + 3iT^{2} \)
7 \( 1 + (-2.09 - 2.09i)T + 7iT^{2} \)
11 \( 1 + 1.39iT - 11T^{2} \)
13 \( 1 + (-4.24 - 4.24i)T + 13iT^{2} \)
17 \( 1 + (4.38 + 4.38i)T + 17iT^{2} \)
19 \( 1 + 2.37T + 19T^{2} \)
29 \( 1 - 3.87iT - 29T^{2} \)
31 \( 1 + 5.74T + 31T^{2} \)
37 \( 1 + (2.27 + 2.27i)T + 37iT^{2} \)
41 \( 1 - 2.49T + 41T^{2} \)
43 \( 1 + (2.85 - 2.85i)T - 43iT^{2} \)
47 \( 1 + (1.26 - 1.26i)T - 47iT^{2} \)
53 \( 1 + (-4.13 + 4.13i)T - 53iT^{2} \)
59 \( 1 + 5.66iT - 59T^{2} \)
61 \( 1 - 13.1iT - 61T^{2} \)
67 \( 1 + (11.1 + 11.1i)T + 67iT^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 + (-10.7 - 10.7i)T + 73iT^{2} \)
79 \( 1 + 5.70T + 79T^{2} \)
83 \( 1 + (-11.6 + 11.6i)T - 83iT^{2} \)
89 \( 1 - 1.31T + 89T^{2} \)
97 \( 1 + (-7.84 - 7.84i)T + 97iT^{2} \)
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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

−11.90889539715936109521829860200, −11.32381214397072705318629282090, −10.61186192375345110846307808967, −9.176848498722289623250048577860, −8.663142776225284780425198678980, −6.90031748097711714709150163755, −6.06638786962809521131402961296, −4.54828264481667800641502026230, −3.44074537551477681456667542450, −2.18026424700843169557694441266, 1.71284667755214513593035269803, 3.84951529013848281097956935373, 4.82645011404514831854181383839, 5.91194337050684054918545524439, 7.41954392408700519536750137265, 8.126506102310104290892621361512, 8.801345022133054394412383706577, 10.48189253204556179855115207865, 11.26435552072301645629332382485, 12.62746786959135580503668397328

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