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Properties

Label 2-230-115.114-c2-0-18
Degree $2$
Conductor $230$
Sign $-0.727 + 0.685i$
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 + 5.56i·3-s − 2.00·4-s + (0.637 − 4.95i)5-s + 7.87·6-s − 10.0·7-s + 2.82i·8-s − 22.0·9-s + (−7.01 − 0.901i)10-s − 9.80i·11-s − 11.1i·12-s − 14.3i·13-s + 14.1i·14-s + (27.6 + 3.54i)15-s + 4.00·16-s + 17.3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.85i·3-s − 0.500·4-s + (0.127 − 0.991i)5-s + 1.31·6-s − 1.43·7-s + 0.353i·8-s − 2.44·9-s + (−0.701 − 0.0901i)10-s − 0.891i·11-s − 0.928i·12-s − 1.10i·13-s + 1.01i·14-s + (1.84 + 0.236i)15-s + 0.250·16-s + 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.727 + 0.685i$
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: \((2,\ 230,\ (\ :1),\ -0.727 + 0.685i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.150741 - 0.379797i\)
\(L(\frac12)\) \(\approx\) \(0.150741 - 0.379797i\)
\(L(2)\) 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 \( 1 + (-0.637 + 4.95i)T \)
23 \( 1 + (14.5 - 17.7i)T \)
good3 \( 1 - 5.56iT - 9T^{2} \)
7 \( 1 + 10.0T + 49T^{2} \)
11 \( 1 + 9.80iT - 121T^{2} \)
13 \( 1 + 14.3iT - 169T^{2} \)
17 \( 1 - 17.3T + 289T^{2} \)
19 \( 1 + 8.38iT - 361T^{2} \)
29 \( 1 + 26.9T + 841T^{2} \)
31 \( 1 - 4.24T + 961T^{2} \)
37 \( 1 + 73.2T + 1.36e3T^{2} \)
41 \( 1 + 19.8T + 1.68e3T^{2} \)
43 \( 1 - 18.1T + 1.84e3T^{2} \)
47 \( 1 - 10.2iT - 2.20e3T^{2} \)
53 \( 1 + 90.6T + 2.80e3T^{2} \)
59 \( 1 - 92.9T + 3.48e3T^{2} \)
61 \( 1 + 36.3iT - 3.72e3T^{2} \)
67 \( 1 - 41.4T + 4.48e3T^{2} \)
71 \( 1 + 22.7T + 5.04e3T^{2} \)
73 \( 1 - 45.3iT - 5.32e3T^{2} \)
79 \( 1 - 65.6iT - 6.24e3T^{2} \)
83 \( 1 - 4.65T + 6.88e3T^{2} \)
89 \( 1 + 121. iT - 7.92e3T^{2} \)
97 \( 1 - 99.8T + 9.40e3T^{2} \)
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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

−11.43187994298774355951444374787, −10.36497073580719602318154279972, −9.792728504238044976781059794593, −9.130742868496736403130183829238, −8.220558441551750436840378394148, −5.80084638096110528980166669974, −5.20825968980841756526867881241, −3.73837456989385979950809637441, −3.19123108970266382299096123020, −0.20965162537681345256364203097, 2.01200726397427016791928244555, 3.44760713810492014285681015218, 5.76616783370683689200650549174, 6.65623680773235197909888704229, 6.99401712221387613126149051556, 7.935161731370767578406253901675, 9.266916793770619673967341589127, 10.28207116693934087031084618212, 11.85530603781619090100381968533, 12.46236945527316795339159236460

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