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Properties

Label 2-230-5.4-c5-0-22
Degree $2$
Conductor $230$
Sign $0.335 - 0.942i$
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 + 19.5i·3-s − 16·4-s + (−18.7 + 52.6i)5-s + 78.1·6-s − 153. i·7-s + 64i·8-s − 138.·9-s + (210. + 75.0i)10-s + 702.·11-s − 312. i·12-s + 760. i·13-s − 612.·14-s + (−1.02e3 − 366. i)15-s + 256·16-s − 1.46e3i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.25i·3-s − 0.5·4-s + (−0.335 + 0.942i)5-s + 0.886·6-s − 1.18i·7-s + 0.353i·8-s − 0.570·9-s + (0.666 + 0.237i)10-s + 1.74·11-s − 0.626i·12-s + 1.24i·13-s − 0.834·14-s + (−1.18 − 0.420i)15-s + 0.250·16-s − 1.23i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.335 - 0.942i$
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: \((2,\ 230,\ (\ :5/2),\ 0.335 - 0.942i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.897110786\)
\(L(\frac12)\) \(\approx\) \(1.897110786\)
\(L(\frac{7}{2})\) 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 \( 1 + (18.7 - 52.6i)T \)
23 \( 1 - 529iT \)
good3 \( 1 - 19.5iT - 243T^{2} \)
7 \( 1 + 153. iT - 1.68e4T^{2} \)
11 \( 1 - 702.T + 1.61e5T^{2} \)
13 \( 1 - 760. iT - 3.71e5T^{2} \)
17 \( 1 + 1.46e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.89e3T + 2.47e6T^{2} \)
29 \( 1 - 221.T + 2.05e7T^{2} \)
31 \( 1 - 2.89e3T + 2.86e7T^{2} \)
37 \( 1 - 705. iT - 6.93e7T^{2} \)
41 \( 1 + 1.36e4T + 1.15e8T^{2} \)
43 \( 1 - 1.35e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.25e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.06e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.89e4T + 7.14e8T^{2} \)
61 \( 1 + 1.42e4T + 8.44e8T^{2} \)
67 \( 1 - 1.00e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.46e4T + 1.80e9T^{2} \)
73 \( 1 + 5.37e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.82e4T + 3.07e9T^{2} \)
83 \( 1 - 2.32e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.27e4T + 5.58e9T^{2} \)
97 \( 1 - 5.96e4iT - 8.58e9T^{2} \)
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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

−11.45767351914476235623722116877, −10.58560891804326724574198024641, −9.638826173942347219825831407487, −9.237934450821046121630777353971, −7.45514957054372413933324848599, −6.55849074789531597437907913510, −4.65132479710868351792863066716, −3.98338611894851939876391525392, −3.16327273993716529181984192378, −1.19464224605983015785606579469, 0.67527849094477873694387710558, 1.72487761413333777473669427602, 3.66815852734666380791537493998, 5.26123024513122625224276402108, 6.10312452862863412955501515953, 7.09777897350363184555628344458, 8.279542758018776389096370729700, 8.659667089382478198948212419915, 9.842775897505826099817259038292, 11.77945016386025535946714440444

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