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Properties

Label 2-230-115.114-c2-0-14
Degree $2$
Conductor $230$
Sign $0.386 + 0.922i$
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  − 1.41i·2-s − 2.29i·3-s − 2.00·4-s + (2.28 + 4.44i)5-s − 3.23·6-s + 9.92·7-s + 2.82i·8-s + 3.75·9-s + (6.29 − 3.22i)10-s − 3.77i·11-s + 4.58i·12-s + 6.77i·13-s − 14.0i·14-s + (10.1 − 5.22i)15-s + 4.00·16-s + 8.36·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.763i·3-s − 0.500·4-s + (0.456 + 0.889i)5-s − 0.539·6-s + 1.41·7-s + 0.353i·8-s + 0.417·9-s + (0.629 − 0.322i)10-s − 0.342i·11-s + 0.381i·12-s + 0.521i·13-s − 1.00i·14-s + (0.679 − 0.348i)15-s + 0.250·16-s + 0.491·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.57846 - 1.04963i\)
\(L(\frac12)\) \(\approx\) \(1.57846 - 1.04963i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
5 \( 1 + (-2.28 - 4.44i)T \)
23 \( 1 + (-1.76 + 22.9i)T \)
good3 \( 1 + 2.29iT - 9T^{2} \)
7 \( 1 - 9.92T + 49T^{2} \)
11 \( 1 + 3.77iT - 121T^{2} \)
13 \( 1 - 6.77iT - 169T^{2} \)
17 \( 1 - 8.36T + 289T^{2} \)
19 \( 1 + 3.59iT - 361T^{2} \)
29 \( 1 - 5.85T + 841T^{2} \)
31 \( 1 + 42.7T + 961T^{2} \)
37 \( 1 + 4.28T + 1.36e3T^{2} \)
41 \( 1 + 25.3T + 1.68e3T^{2} \)
43 \( 1 - 65.1T + 1.84e3T^{2} \)
47 \( 1 + 5.16iT - 2.20e3T^{2} \)
53 \( 1 + 8.63T + 2.80e3T^{2} \)
59 \( 1 - 82.3T + 3.48e3T^{2} \)
61 \( 1 - 108. iT - 3.72e3T^{2} \)
67 \( 1 - 30.7T + 4.48e3T^{2} \)
71 \( 1 + 117.T + 5.04e3T^{2} \)
73 \( 1 + 81.8iT - 5.32e3T^{2} \)
79 \( 1 - 138. iT - 6.24e3T^{2} \)
83 \( 1 + 33.5T + 6.88e3T^{2} \)
89 \( 1 + 49.1iT - 7.92e3T^{2} \)
97 \( 1 + 138.T + 9.40e3T^{2} \)
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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

−11.69719009799099415626845582631, −10.95879524056788999634469694872, −10.15056160241284813748186781871, −8.872268588033984917991423362627, −7.76997654389223168298876055108, −6.87447295358031661374682217953, −5.53563654293553003278600670498, −4.15628864294948467370450771017, −2.46209047115308927343818509114, −1.40813370408760870059613383182, 1.50514987124522523936604740931, 3.99334208539356941714454158245, 4.99746079619072496131703910570, 5.57946705483053906946738988517, 7.34516740547436683460160328642, 8.206186954165878498467548721345, 9.201160038902052266479412675548, 10.00530375493068452133296502110, 11.05143493148597912623784183591, 12.29454069334549668376307561758

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