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Properties

Label 2-230-5.4-c5-0-23
Degree $2$
Conductor $230$
Sign $-0.142 - 0.989i$
Analytic cond. $36.8882$
Root an. cond. $6.07357$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s + 10.0i·3-s − 16·4-s + (7.97 + 55.3i)5-s + 40.0·6-s + 231. i·7-s + 64i·8-s + 142.·9-s + (221. − 31.9i)10-s + 512.·11-s − 160. i·12-s − 758. i·13-s + 926.·14-s + (−553. + 79.8i)15-s + 256·16-s + 1.48e3i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.641i·3-s − 0.5·4-s + (0.142 + 0.989i)5-s + 0.453·6-s + 1.78i·7-s + 0.353i·8-s + 0.587·9-s + (0.699 − 0.100i)10-s + 1.27·11-s − 0.320i·12-s − 1.24i·13-s + 1.26·14-s + (−0.635 + 0.0915i)15-s + 0.250·16-s + 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.142 - 0.989i$
Analytic conductor: \(36.8882\)
Root analytic conductor: \(6.07357\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :5/2),\ -0.142 - 0.989i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.142455167\)
\(L(\frac12)\) \(\approx\) \(2.142455167\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
5 \( 1 + (-7.97 - 55.3i)T \)
23 \( 1 - 529iT \)
good3 \( 1 - 10.0iT - 243T^{2} \)
7 \( 1 - 231. iT - 1.68e4T^{2} \)
11 \( 1 - 512.T + 1.61e5T^{2} \)
13 \( 1 + 758. iT - 3.71e5T^{2} \)
17 \( 1 - 1.48e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.81e3T + 2.47e6T^{2} \)
29 \( 1 + 7.88e3T + 2.05e7T^{2} \)
31 \( 1 - 6.07e3T + 2.86e7T^{2} \)
37 \( 1 - 9.25e3iT - 6.93e7T^{2} \)
41 \( 1 - 2.08e4T + 1.15e8T^{2} \)
43 \( 1 + 3.94e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.91e4iT - 2.29e8T^{2} \)
53 \( 1 - 515. iT - 4.18e8T^{2} \)
59 \( 1 + 4.23e4T + 7.14e8T^{2} \)
61 \( 1 + 156.T + 8.44e8T^{2} \)
67 \( 1 + 1.85e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.76e4T + 1.80e9T^{2} \)
73 \( 1 - 1.24e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.14e4T + 3.07e9T^{2} \)
83 \( 1 + 4.04e3iT - 3.93e9T^{2} \)
89 \( 1 - 7.93e4T + 5.58e9T^{2} \)
97 \( 1 - 5.58e4iT - 8.58e9T^{2} \)
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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

−11.56618031067789051347003928328, −10.62059276937288385785818308138, −9.711910117730167327669530701689, −9.088875183791744582387047961597, −7.81632104662425182177990048659, −6.24135129734640286439318293792, −5.38221888879584605811048081122, −3.83102655173735146793915587147, −2.92353753271859659230488933797, −1.61704703917633114686960323385, 0.71161788543120235152365152901, 1.40974368317314602057960069378, 4.01892889094082909173411924995, 4.58785184135246940887906319941, 6.21975243173851956441189856140, 7.24457386569896824549408581717, 7.61147308150574996432358202591, 9.276702369657915928293840106579, 9.605457455626535132240483026986, 11.22925607263479341550502241890

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