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Properties

Label 2-230-115.97-c1-0-7
Degree $2$
Conductor $230$
Sign $0.814 - 0.580i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.479 + 0.877i)2-s + (−0.0548 − 0.767i)3-s + (−0.540 + 0.841i)4-s + (2.20 + 0.341i)5-s + (0.647 − 0.415i)6-s + (−0.484 − 0.180i)7-s + (−0.997 − 0.0713i)8-s + (2.38 − 0.342i)9-s + (0.759 + 2.10i)10-s + (0.0487 + 0.166i)11-s + (0.675 + 0.368i)12-s + (0.678 + 1.81i)13-s + (−0.0735 − 0.511i)14-s + (0.140 − 1.71i)15-s + (−0.415 − 0.909i)16-s + (−0.00793 + 0.0364i)17-s + ⋯
L(s)  = 1  + (0.338 + 0.620i)2-s + (−0.0316 − 0.443i)3-s + (−0.270 + 0.420i)4-s + (0.988 + 0.152i)5-s + (0.264 − 0.169i)6-s + (−0.183 − 0.0683i)7-s + (−0.352 − 0.0252i)8-s + (0.794 − 0.114i)9-s + (0.240 + 0.665i)10-s + (0.0147 + 0.0500i)11-s + (0.194 + 0.106i)12-s + (0.188 + 0.504i)13-s + (−0.0196 − 0.136i)14-s + (0.0362 − 0.442i)15-s + (−0.103 − 0.227i)16-s + (−0.00192 + 0.00884i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.814 - 0.580i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.814 - 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54351 + 0.494141i\)
\(L(\frac12)\) \(\approx\) \(1.54351 + 0.494141i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.479 - 0.877i)T \)
5 \( 1 + (-2.20 - 0.341i)T \)
23 \( 1 + (4.61 - 1.29i)T \)
good3 \( 1 + (0.0548 + 0.767i)T + (-2.96 + 0.426i)T^{2} \)
7 \( 1 + (0.484 + 0.180i)T + (5.29 + 4.58i)T^{2} \)
11 \( 1 + (-0.0487 - 0.166i)T + (-9.25 + 5.94i)T^{2} \)
13 \( 1 + (-0.678 - 1.81i)T + (-9.82 + 8.51i)T^{2} \)
17 \( 1 + (0.00793 - 0.0364i)T + (-15.4 - 7.06i)T^{2} \)
19 \( 1 + (-0.624 - 0.401i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (1.50 + 2.34i)T + (-12.0 + 26.3i)T^{2} \)
31 \( 1 + (1.86 + 2.15i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (6.38 + 4.78i)T + (10.4 + 35.5i)T^{2} \)
41 \( 1 + (0.673 - 4.68i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (6.52 - 0.466i)T + (42.5 - 6.11i)T^{2} \)
47 \( 1 + (5.54 + 5.54i)T + 47iT^{2} \)
53 \( 1 + (0.476 - 1.27i)T + (-40.0 - 34.7i)T^{2} \)
59 \( 1 + (-2.12 - 0.970i)T + (38.6 + 44.5i)T^{2} \)
61 \( 1 + (-6.29 + 5.45i)T + (8.68 - 60.3i)T^{2} \)
67 \( 1 + (-0.468 + 0.255i)T + (36.2 - 56.3i)T^{2} \)
71 \( 1 + (-13.0 - 3.83i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (3.75 - 0.816i)T + (66.4 - 30.3i)T^{2} \)
79 \( 1 + (-0.561 + 1.22i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (6.52 - 8.72i)T + (-23.3 - 79.6i)T^{2} \)
89 \( 1 + (-0.385 + 0.445i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-3.69 - 4.94i)T + (-27.3 + 93.0i)T^{2} \)
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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

−12.62156323255684720033657851632, −11.53638034382591792735068640711, −10.13122441108074131032177946400, −9.446584585080083035873678663682, −8.169174376086996989931776653292, −6.99591321614480477028479503056, −6.33752286721967665819134795102, −5.21159553162139380800315083157, −3.78549163744496863536561037859, −1.92478398552814351900867555559, 1.72721378667716379610262672476, 3.32218739405990878086843642782, 4.69497538815370141844143736824, 5.65597684371251492243618541530, 6.84393136168649928348247255892, 8.458907585492090458764948211311, 9.595252135277737391547547290665, 10.15602226234238638500213735349, 10.97207661265246868055668060270, 12.28106572052364566641092033840

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