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Properties

Label 2-230-5.2-c4-0-25
Degree $2$
Conductor $230$
Sign $0.597 + 0.801i$
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 − 2i)2-s + (−2.20 + 2.20i)3-s + 8i·4-s + (−2.16 − 24.9i)5-s + 8.82·6-s + (−2.30 − 2.30i)7-s + (16 − 16i)8-s + 71.2i·9-s + (−45.4 + 54.1i)10-s + 77.3·11-s + (−17.6 − 17.6i)12-s + (15.2 − 15.2i)13-s + 9.22i·14-s + (59.7 + 50.1i)15-s − 64·16-s + (−132. − 132. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.245 + 0.245i)3-s + 0.5i·4-s + (−0.0867 − 0.996i)5-s + 0.245·6-s + (−0.0470 − 0.0470i)7-s + (0.250 − 0.250i)8-s + 0.879i·9-s + (−0.454 + 0.541i)10-s + 0.639·11-s + (−0.122 − 0.122i)12-s + (0.0903 − 0.0903i)13-s + 0.0470i·14-s + (0.265 + 0.222i)15-s − 0.250·16-s + (−0.459 − 0.459i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.597 + 0.801i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.597 + 0.801i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.252342951\)
\(L(\frac12)\) \(\approx\) \(1.252342951\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
5 \( 1 + (2.16 + 24.9i)T \)
23 \( 1 + (-77.9 + 77.9i)T \)
good3 \( 1 + (2.20 - 2.20i)T - 81iT^{2} \)
7 \( 1 + (2.30 + 2.30i)T + 2.40e3iT^{2} \)
11 \( 1 - 77.3T + 1.46e4T^{2} \)
13 \( 1 + (-15.2 + 15.2i)T - 2.85e4iT^{2} \)
17 \( 1 + (132. + 132. i)T + 8.35e4iT^{2} \)
19 \( 1 - 569. iT - 1.30e5T^{2} \)
29 \( 1 + 1.14e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.56e3T + 9.23e5T^{2} \)
37 \( 1 + (-1.81e3 - 1.81e3i)T + 1.87e6iT^{2} \)
41 \( 1 - 211.T + 2.82e6T^{2} \)
43 \( 1 + (-1.73e3 + 1.73e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (784. + 784. i)T + 4.87e6iT^{2} \)
53 \( 1 + (-3.44e3 + 3.44e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 1.09e3iT - 1.21e7T^{2} \)
61 \( 1 - 54.6T + 1.38e7T^{2} \)
67 \( 1 + (4.19e3 + 4.19e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 5.73e3T + 2.54e7T^{2} \)
73 \( 1 + (-2.95e3 + 2.95e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 3.54e3iT - 3.89e7T^{2} \)
83 \( 1 + (-5.30e3 + 5.30e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 771. iT - 6.27e7T^{2} \)
97 \( 1 + (-7.95e3 - 7.95e3i)T + 8.85e7iT^{2} \)
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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

−11.54266725249142961769679860751, −10.28490011942236554464660040087, −9.621526397206508582884340600332, −8.454394141317077016464451224024, −7.82039739209565772480026023255, −6.22036239767135682104519637305, −4.90163737855061303849075632503, −3.93062005996542351658370317076, −2.12120305568451886781952201962, −0.72803691780295146890350751183, 0.917176772751631757251984013102, 2.75081162227491660228429745901, 4.26947296792250543907818914428, 5.99425229978034313484031098404, 6.67249331064017462296136875364, 7.44000159067846851659998232388, 8.845939330650705728664204954111, 9.564510725402786962900916680373, 10.82364475553804738036865323494, 11.42074456307719668098598167017

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