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Properties

Label 2-230-23.22-c4-0-21
Degree $2$
Conductor $230$
Sign $-0.928 + 0.372i$
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  + 2.82·2-s − 16.9·3-s + 8.00·4-s + 11.1i·5-s − 48.0·6-s − 34.8i·7-s + 22.6·8-s + 207.·9-s + 31.6i·10-s + 184. i·11-s − 135.·12-s + 3.47·13-s − 98.5i·14-s − 189. i·15-s + 64.0·16-s − 104. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.88·3-s + 0.500·4-s + 0.447i·5-s − 1.33·6-s − 0.711i·7-s + 0.353·8-s + 2.55·9-s + 0.316i·10-s + 1.52i·11-s − 0.943·12-s + 0.0205·13-s − 0.502i·14-s − 0.843i·15-s + 0.250·16-s − 0.361i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.372i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.928 + 0.372i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.928 + 0.372i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1250012568\)
\(L(\frac12)\) \(\approx\) \(0.1250012568\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82T \)
5 \( 1 - 11.1iT \)
23 \( 1 + (197. + 490. i)T \)
good3 \( 1 + 16.9T + 81T^{2} \)
7 \( 1 + 34.8iT - 2.40e3T^{2} \)
11 \( 1 - 184. iT - 1.46e4T^{2} \)
13 \( 1 - 3.47T + 2.85e4T^{2} \)
17 \( 1 + 104. iT - 8.35e4T^{2} \)
19 \( 1 - 8.08iT - 1.30e5T^{2} \)
29 \( 1 + 694.T + 7.07e5T^{2} \)
31 \( 1 + 1.47e3T + 9.23e5T^{2} \)
37 \( 1 - 89.3iT - 1.87e6T^{2} \)
41 \( 1 + 1.01e3T + 2.82e6T^{2} \)
43 \( 1 + 3.42e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.76e3T + 4.87e6T^{2} \)
53 \( 1 - 3.85e3iT - 7.89e6T^{2} \)
59 \( 1 + 5.48e3T + 1.21e7T^{2} \)
61 \( 1 + 4.33e3iT - 1.38e7T^{2} \)
67 \( 1 - 2.33e3iT - 2.01e7T^{2} \)
71 \( 1 + 3.60e3T + 2.54e7T^{2} \)
73 \( 1 + 208.T + 2.83e7T^{2} \)
79 \( 1 - 2.06e3iT - 3.89e7T^{2} \)
83 \( 1 - 8.41e3iT - 4.74e7T^{2} \)
89 \( 1 - 2.13e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.54e4iT - 8.85e7T^{2} \)
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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

−11.13892495382070742271454951024, −10.54455614752331888227022147557, −9.732536659661664531571808750951, −7.32444022787683007471733478060, −6.96642654311202543673805113699, −5.84383803392896553365634256464, −4.84202795287886198903702609718, −3.99901974541303851866963548175, −1.76916889472187892691741692474, −0.04290431833456102285872579731, 1.47465841882150757074522505320, 3.66771298261890425696272182663, 5.02768474107195662840413364263, 5.73582964686696502121160191343, 6.31024578836735228502644249082, 7.70548589279705614652827519540, 9.200415157958163228250500819329, 10.50791354785451524254618572267, 11.38099739393231645487892503246, 11.77234134447581638094539854209

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