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Properties

Label 2-230-5.4-c3-0-8
Degree $2$
Conductor $230$
Sign $-0.396 + 0.918i$
Analytic cond. $13.5704$
Root an. cond. $3.68380$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 7.80i·3-s − 4·4-s + (−4.43 + 10.2i)5-s − 15.6·6-s + 21.1i·7-s − 8i·8-s − 33.9·9-s + (−20.5 − 8.86i)10-s + 4.34·11-s − 31.2i·12-s + 43.4i·13-s − 42.3·14-s + (−80.1 − 34.6i)15-s + 16·16-s − 0.244i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.50i·3-s − 0.5·4-s + (−0.396 + 0.918i)5-s − 1.06·6-s + 1.14i·7-s − 0.353i·8-s − 1.25·9-s + (−0.649 − 0.280i)10-s + 0.119·11-s − 0.751i·12-s + 0.926i·13-s − 0.807·14-s + (−1.37 − 0.595i)15-s + 0.250·16-s − 0.00349i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.396 + 0.918i$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ -0.396 + 0.918i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.755750 - 1.14974i\)
\(L(\frac12)\) \(\approx\) \(0.755750 - 1.14974i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (4.43 - 10.2i)T \)
23 \( 1 + 23iT \)
good3 \( 1 - 7.80iT - 27T^{2} \)
7 \( 1 - 21.1iT - 343T^{2} \)
11 \( 1 - 4.34T + 1.33e3T^{2} \)
13 \( 1 - 43.4iT - 2.19e3T^{2} \)
17 \( 1 + 0.244iT - 4.91e3T^{2} \)
19 \( 1 - 141.T + 6.85e3T^{2} \)
29 \( 1 - 91.2T + 2.43e4T^{2} \)
31 \( 1 - 79.7T + 2.97e4T^{2} \)
37 \( 1 - 56.7iT - 5.06e4T^{2} \)
41 \( 1 - 384.T + 6.89e4T^{2} \)
43 \( 1 + 297. iT - 7.95e4T^{2} \)
47 \( 1 + 69.2iT - 1.03e5T^{2} \)
53 \( 1 + 237. iT - 1.48e5T^{2} \)
59 \( 1 - 234.T + 2.05e5T^{2} \)
61 \( 1 + 938.T + 2.26e5T^{2} \)
67 \( 1 - 31.0iT - 3.00e5T^{2} \)
71 \( 1 + 995.T + 3.57e5T^{2} \)
73 \( 1 - 807. iT - 3.89e5T^{2} \)
79 \( 1 + 410.T + 4.93e5T^{2} \)
83 \( 1 - 287. iT - 5.71e5T^{2} \)
89 \( 1 - 1.24e3T + 7.04e5T^{2} \)
97 \( 1 - 1.41e3iT - 9.12e5T^{2} \)
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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

−12.08576113870439603602682825154, −11.40827192378456991901840312759, −10.27997449325336302476239617274, −9.447331594759316696239435095186, −8.715287870000887863972968920020, −7.40997756947473068162579731031, −6.17577597112174242966019841705, −5.14299539094342394902562109524, −4.03889649283400677213026669376, −2.87941180580225039374219773194, 0.65745135156391597569051978304, 1.30028673500394124075377874972, 3.11487397558523962846994299465, 4.56355032215327709575502864504, 5.92110778171509331898473630134, 7.50845144918712294430362150025, 7.77990621517204298379963739466, 9.079880522245175083665891132346, 10.25575581420062804716461319341, 11.43767072117011732280827639949

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