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Properties

Label 2-230-23.22-c2-0-9
Degree $2$
Conductor $230$
Sign $0.963 + 0.266i$
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 + 4.76·3-s + 2.00·4-s + 2.23i·5-s − 6.73·6-s − 7.05i·7-s − 2.82·8-s + 13.6·9-s − 3.16i·10-s − 10.4i·11-s + 9.52·12-s + 19.0·13-s + 9.98i·14-s + 10.6i·15-s + 4.00·16-s − 12.8i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.58·3-s + 0.500·4-s + 0.447i·5-s − 1.12·6-s − 1.00i·7-s − 0.353·8-s + 1.52·9-s − 0.316i·10-s − 0.950i·11-s + 0.793·12-s + 1.46·13-s + 0.713i·14-s + 0.710i·15-s + 0.250·16-s − 0.754i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.963 + 0.266i$
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: \((2,\ 230,\ (\ :1),\ 0.963 + 0.266i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.95112 - 0.265268i\)
\(L(\frac12)\) \(\approx\) \(1.95112 - 0.265268i\)
\(L(2)\) 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 \( 1 + (6.14 - 22.1i)T \)
good3 \( 1 - 4.76T + 9T^{2} \)
7 \( 1 + 7.05iT - 49T^{2} \)
11 \( 1 + 10.4iT - 121T^{2} \)
13 \( 1 - 19.0T + 169T^{2} \)
17 \( 1 + 12.8iT - 289T^{2} \)
19 \( 1 - 22.7iT - 361T^{2} \)
29 \( 1 + 8.61T + 841T^{2} \)
31 \( 1 - 22.2T + 961T^{2} \)
37 \( 1 - 29.8iT - 1.36e3T^{2} \)
41 \( 1 + 18.7T + 1.68e3T^{2} \)
43 \( 1 + 10.0iT - 1.84e3T^{2} \)
47 \( 1 - 20.6T + 2.20e3T^{2} \)
53 \( 1 + 17.3iT - 2.80e3T^{2} \)
59 \( 1 + 103.T + 3.48e3T^{2} \)
61 \( 1 + 74.6iT - 3.72e3T^{2} \)
67 \( 1 - 101. iT - 4.48e3T^{2} \)
71 \( 1 + 69.4T + 5.04e3T^{2} \)
73 \( 1 + 122.T + 5.32e3T^{2} \)
79 \( 1 + 140. iT - 6.24e3T^{2} \)
83 \( 1 - 126. iT - 6.88e3T^{2} \)
89 \( 1 + 11.9iT - 7.92e3T^{2} \)
97 \( 1 + 57.0iT - 9.40e3T^{2} \)
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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

−11.68686099902551232922680423066, −10.67870605397094859143784296153, −9.881593793037392720305399364015, −8.859067027069503842118417775467, −8.090417241076185757696397639288, −7.36285676960141957540610527727, −6.11926800976995905851383602723, −3.83137760550846842473894954644, −3.11524104739889098183972782282, −1.40769596294195916665714653287, 1.74223207996868008990645008745, 2.84675328175601626483795115355, 4.30812574053747750970919532207, 6.08290410763918488886207924221, 7.40799569191094415851231731673, 8.563020165695860311471399072175, 8.768015414159490959294133561782, 9.674566165075280998243400649229, 10.82238406737215710490927663507, 12.15197638791110547689180715956

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