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Properties

Label 2-230-115.114-c4-0-39
Degree $2$
Conductor $230$
Sign $-0.878 + 0.478i$
Analytic cond. $23.7750$
Root an. cond. $4.87597$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82i·2-s − 1.19i·3-s − 8.00·4-s + (−1.65 − 24.9i)5-s − 3.37·6-s + 28.6·7-s + 22.6i·8-s + 79.5·9-s + (−70.5 + 4.69i)10-s − 13.9i·11-s + 9.54i·12-s − 186. i·13-s − 81.1i·14-s + (−29.7 + 1.98i)15-s + 64.0·16-s + 309.·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.132i·3-s − 0.500·4-s + (−0.0663 − 0.997i)5-s − 0.0937·6-s + 0.585·7-s + 0.353i·8-s + 0.982·9-s + (−0.705 + 0.0469i)10-s − 0.115i·11-s + 0.0662i·12-s − 1.10i·13-s − 0.414i·14-s + (−0.132 + 0.00880i)15-s + 0.250·16-s + 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.478i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.878 + 0.478i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.878 + 0.478i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.862411804\)
\(L(\frac12)\) \(\approx\) \(1.862411804\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
5 \( 1 + (1.65 + 24.9i)T \)
23 \( 1 + (-480. + 221. i)T \)
good3 \( 1 + 1.19iT - 81T^{2} \)
7 \( 1 - 28.6T + 2.40e3T^{2} \)
11 \( 1 + 13.9iT - 1.46e4T^{2} \)
13 \( 1 + 186. iT - 2.85e4T^{2} \)
17 \( 1 - 309.T + 8.35e4T^{2} \)
19 \( 1 + 55.0iT - 1.30e5T^{2} \)
29 \( 1 + 973.T + 7.07e5T^{2} \)
31 \( 1 + 382.T + 9.23e5T^{2} \)
37 \( 1 + 1.34e3T + 1.87e6T^{2} \)
41 \( 1 + 103.T + 2.82e6T^{2} \)
43 \( 1 - 1.37e3T + 3.41e6T^{2} \)
47 \( 1 + 1.76e3iT - 4.87e6T^{2} \)
53 \( 1 - 783.T + 7.89e6T^{2} \)
59 \( 1 + 3.56e3T + 1.21e7T^{2} \)
61 \( 1 + 4.45e3iT - 1.38e7T^{2} \)
67 \( 1 - 227.T + 2.01e7T^{2} \)
71 \( 1 + 5.19e3T + 2.54e7T^{2} \)
73 \( 1 + 704. iT - 2.83e7T^{2} \)
79 \( 1 - 5.46e3iT - 3.89e7T^{2} \)
83 \( 1 + 363.T + 4.74e7T^{2} \)
89 \( 1 - 7.68e3iT - 6.27e7T^{2} \)
97 \( 1 - 3.73e3T + 8.85e7T^{2} \)
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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

−11.15677293321482887136597065436, −10.22484751532401774348773595003, −9.287189704166534759874002566615, −8.254183118935642167807640671624, −7.40356698583748047504869446884, −5.56843686291450253607415275631, −4.73458670897168171592420661381, −3.48176512072658568944867876356, −1.71675843075533415542148541342, −0.68138921685379643951223845826, 1.63724685999066850646582927439, 3.51265819585067857003905553706, 4.64269759194664893763509225442, 5.92898117597905781048828897909, 7.15873462589153943404753445766, 7.56156519028811372626028378533, 9.032456297181801343887666101943, 9.945256585353661622436020087322, 10.90437044313736271488285591506, 11.88491305470795980253643670433

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