Properties

Label 2-230-23.22-c2-0-14
Degree $2$
Conductor $230$
Sign $-0.196 + 0.980i$
Analytic cond. $6.26704$
Root an. cond. $2.50340$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 2.34·3-s + 2.00·4-s − 2.23i·5-s − 3.32·6-s − 7.61i·7-s − 2.82·8-s − 3.48·9-s + 3.16i·10-s − 12.3i·11-s + 4.69·12-s − 13.0·13-s + 10.7i·14-s − 5.25i·15-s + 4.00·16-s + 9.13i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.782·3-s + 0.500·4-s − 0.447i·5-s − 0.553·6-s − 1.08i·7-s − 0.353·8-s − 0.387·9-s + 0.316i·10-s − 1.12i·11-s + 0.391·12-s − 1.00·13-s + 0.769i·14-s − 0.350i·15-s + 0.250·16-s + 0.537i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.196 + 0.980i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ -0.196 + 0.980i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.714893 - 0.872117i\)
\(L(\frac12)\) \(\approx\) \(0.714893 - 0.872117i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
5 \( 1 + 2.23iT \)
23 \( 1 + (-22.5 - 4.51i)T \)
good3 \( 1 - 2.34T + 9T^{2} \)
7 \( 1 + 7.61iT - 49T^{2} \)
11 \( 1 + 12.3iT - 121T^{2} \)
13 \( 1 + 13.0T + 169T^{2} \)
17 \( 1 - 9.13iT - 289T^{2} \)
19 \( 1 + 14.4iT - 361T^{2} \)
29 \( 1 + 21.2T + 841T^{2} \)
31 \( 1 - 36.8T + 961T^{2} \)
37 \( 1 + 56.9iT - 1.36e3T^{2} \)
41 \( 1 - 70.7T + 1.68e3T^{2} \)
43 \( 1 + 70.0iT - 1.84e3T^{2} \)
47 \( 1 + 66.2T + 2.20e3T^{2} \)
53 \( 1 - 77.4iT - 2.80e3T^{2} \)
59 \( 1 - 82.7T + 3.48e3T^{2} \)
61 \( 1 + 23.9iT - 3.72e3T^{2} \)
67 \( 1 - 118. iT - 4.48e3T^{2} \)
71 \( 1 - 69.0T + 5.04e3T^{2} \)
73 \( 1 - 25.9T + 5.32e3T^{2} \)
79 \( 1 - 28.8iT - 6.24e3T^{2} \)
83 \( 1 - 69.3iT - 6.88e3T^{2} \)
89 \( 1 - 45.4iT - 7.92e3T^{2} \)
97 \( 1 - 74.4iT - 9.40e3T^{2} \)
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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

−11.42166760132829842494278533710, −10.69531832109075987274914537694, −9.529874793038411869090611746749, −8.781610152080871371757392174845, −7.902564368744093460844183877658, −7.01915321013879639922811813499, −5.53645714447016920743884654417, −3.91355915766085027960657942585, −2.59311800494650552640840733267, −0.67572871688110102557722460161, 2.17187037121509889412710688680, 3.00678967133823100907823966507, 4.97186457733432161648537851834, 6.38004714735594358980034359398, 7.53739074266405340411690991962, 8.341026540240151101968687302248, 9.442150832456844805467721174003, 9.876443229820738402570139722856, 11.32969440615218131885352704869, 12.07556839894341990045453757731

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