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Properties

Label 2-230-5.4-c5-0-52
Degree $2$
Conductor $230$
Sign $-0.843 - 0.537i$
Analytic cond. $36.8882$
Root an. cond. $6.07357$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 18.6i·3-s − 16·4-s + (−47.1 − 30.0i)5-s + 74.6·6-s − 68.7i·7-s − 64i·8-s − 105.·9-s + (120. − 188. i)10-s − 254.·11-s + 298. i·12-s − 852. i·13-s + 275.·14-s + (−560. + 879. i)15-s + 256·16-s − 1.63e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.19i·3-s − 0.5·4-s + (−0.843 − 0.537i)5-s + 0.846·6-s − 0.530i·7-s − 0.353i·8-s − 0.432·9-s + (0.379 − 0.596i)10-s − 0.635·11-s + 0.598i·12-s − 1.39i·13-s + 0.375·14-s + (−0.642 + 1.00i)15-s + 0.250·16-s − 1.36i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.843 - 0.537i$
Analytic conductor: \(36.8882\)
Root analytic conductor: \(6.07357\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :5/2),\ -0.843 - 0.537i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5169034448\)
\(L(\frac12)\) \(\approx\) \(0.5169034448\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 + (47.1 + 30.0i)T \)
23 \( 1 - 529iT \)
good3 \( 1 + 18.6iT - 243T^{2} \)
7 \( 1 + 68.7iT - 1.68e4T^{2} \)
11 \( 1 + 254.T + 1.61e5T^{2} \)
13 \( 1 + 852. iT - 3.71e5T^{2} \)
17 \( 1 + 1.63e3iT - 1.41e6T^{2} \)
19 \( 1 - 146.T + 2.47e6T^{2} \)
29 \( 1 - 120.T + 2.05e7T^{2} \)
31 \( 1 + 4.52e3T + 2.86e7T^{2} \)
37 \( 1 + 990. iT - 6.93e7T^{2} \)
41 \( 1 + 8.12e3T + 1.15e8T^{2} \)
43 \( 1 - 2.32e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.14e4iT - 2.29e8T^{2} \)
53 \( 1 - 4.07e3iT - 4.18e8T^{2} \)
59 \( 1 + 8.19e3T + 7.14e8T^{2} \)
61 \( 1 - 7.78e3T + 8.44e8T^{2} \)
67 \( 1 - 2.50e3iT - 1.35e9T^{2} \)
71 \( 1 + 3.00e3T + 1.80e9T^{2} \)
73 \( 1 - 1.94e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.43e4T + 3.07e9T^{2} \)
83 \( 1 + 8.13e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.12e5T + 5.58e9T^{2} \)
97 \( 1 + 8.32e4iT - 8.58e9T^{2} \)
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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

−10.89404471313826021200839844139, −9.584862794238042924629985396404, −8.252239988374586678889305934750, −7.63826676330039085527915919932, −7.07905652152796599554637639827, −5.68216477738489902169309506612, −4.61019189303204778151240889716, −3.04890625056892451147682211667, −1.05690528069176005605479152300, −0.17862442804171065063825219045, 2.08416484117968355972606755040, 3.54599231129185788626557603204, 4.19243622931185832426542144673, 5.35868648322823370319677761559, 6.90985183978186835779400322247, 8.341504532892527727490943244537, 9.139142383378147766245825200577, 10.25717263981251017983210661681, 10.79663101211513787902015310592, 11.71454160291804260636595669326

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