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Properties

Label 2-230-115.114-c4-0-32
Degree $2$
Conductor $230$
Sign $0.871 + 0.490i$
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 + 13.4i·3-s − 8.00·4-s + (20.1 − 14.7i)5-s + 38.1·6-s + 78.6·7-s + 22.6i·8-s − 100.·9-s + (−41.7 − 57.0i)10-s − 67.9i·11-s − 107. i·12-s − 229. i·13-s − 222. i·14-s + (198. + 271. i)15-s + 64.0·16-s + 80.2·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.49i·3-s − 0.500·4-s + (0.807 − 0.590i)5-s + 1.05·6-s + 1.60·7-s + 0.353i·8-s − 1.24·9-s + (−0.417 − 0.570i)10-s − 0.561i·11-s − 0.748i·12-s − 1.35i·13-s − 1.13i·14-s + (0.883 + 1.20i)15-s + 0.250·16-s + 0.277·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.871 + 0.490i$
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: \((2,\ 230,\ (\ :2),\ 0.871 + 0.490i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.484026661\)
\(L(\frac12)\) \(\approx\) \(2.484026661\)
\(L(3)\) 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 \( 1 + (-20.1 + 14.7i)T \)
23 \( 1 + (481. - 219. i)T \)
good3 \( 1 - 13.4iT - 81T^{2} \)
7 \( 1 - 78.6T + 2.40e3T^{2} \)
11 \( 1 + 67.9iT - 1.46e4T^{2} \)
13 \( 1 + 229. iT - 2.85e4T^{2} \)
17 \( 1 - 80.2T + 8.35e4T^{2} \)
19 \( 1 + 377. iT - 1.30e5T^{2} \)
29 \( 1 - 1.48e3T + 7.07e5T^{2} \)
31 \( 1 + 1.33e3T + 9.23e5T^{2} \)
37 \( 1 - 975.T + 1.87e6T^{2} \)
41 \( 1 - 2.82e3T + 2.82e6T^{2} \)
43 \( 1 + 748.T + 3.41e6T^{2} \)
47 \( 1 + 2.12e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.43e3T + 7.89e6T^{2} \)
59 \( 1 + 4.87e3T + 1.21e7T^{2} \)
61 \( 1 - 4.85e3iT - 1.38e7T^{2} \)
67 \( 1 - 6.91e3T + 2.01e7T^{2} \)
71 \( 1 + 1.70e3T + 2.54e7T^{2} \)
73 \( 1 - 1.13e3iT - 2.83e7T^{2} \)
79 \( 1 - 7.42e3iT - 3.89e7T^{2} \)
83 \( 1 - 6.14e3T + 4.74e7T^{2} \)
89 \( 1 - 8.56e3iT - 6.27e7T^{2} \)
97 \( 1 + 921.T + 8.85e7T^{2} \)
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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

−11.09283456121755940991569891860, −10.58587573132483019851690446487, −9.704574639919947769066621245357, −8.784073438257338014142549025896, −8.023872779942816889858558077382, −5.59795702619514686138274934512, −5.05237050778494082652244965239, −4.08517999982024215005156540523, −2.60087919976536340482570726246, −0.967695968316892579148911056118, 1.41262124314931507296824645118, 2.15997052166717525363189273626, 4.47379018220289070686943189331, 5.81938544161977732642698293029, 6.64408345773782936221091559745, 7.55261892316756212469225174599, 8.211586732593831285720314117208, 9.457828460387818778617966931143, 10.77276476284796280125000124587, 11.85893552042182867366565215399

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