Properties

Label 2-230-115.64-c1-0-4
Degree $2$
Conductor $230$
Sign $0.931 - 0.364i$
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  + (0.281 − 0.959i)2-s + (1.95 + 1.69i)3-s + (−0.841 − 0.540i)4-s + (−0.542 + 2.16i)5-s + (2.18 − 1.40i)6-s + (−0.748 + 0.341i)7-s + (−0.755 + 0.654i)8-s + (0.529 + 3.68i)9-s + (1.92 + 1.13i)10-s + (4.65 − 1.36i)11-s + (−0.730 − 2.48i)12-s + (−1.40 − 0.639i)13-s + (0.117 + 0.814i)14-s + (−4.74 + 3.32i)15-s + (0.415 + 0.909i)16-s + (−2.11 − 3.28i)17-s + ⋯
L(s)  = 1  + (0.199 − 0.678i)2-s + (1.13 + 0.980i)3-s + (−0.420 − 0.270i)4-s + (−0.242 + 0.970i)5-s + (0.890 − 0.572i)6-s + (−0.282 + 0.129i)7-s + (−0.267 + 0.231i)8-s + (0.176 + 1.22i)9-s + (0.609 + 0.357i)10-s + (1.40 − 0.412i)11-s + (−0.210 − 0.718i)12-s + (−0.388 − 0.177i)13-s + (0.0313 + 0.217i)14-s + (−1.22 + 0.859i)15-s + (0.103 + 0.227i)16-s + (−0.512 − 0.797i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.931 - 0.364i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.931 - 0.364i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70200 + 0.321424i\)
\(L(\frac12)\) \(\approx\) \(1.70200 + 0.321424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.281 + 0.959i)T \)
5 \( 1 + (0.542 - 2.16i)T \)
23 \( 1 + (4.76 + 0.567i)T \)
good3 \( 1 + (-1.95 - 1.69i)T + (0.426 + 2.96i)T^{2} \)
7 \( 1 + (0.748 - 0.341i)T + (4.58 - 5.29i)T^{2} \)
11 \( 1 + (-4.65 + 1.36i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (1.40 + 0.639i)T + (8.51 + 9.82i)T^{2} \)
17 \( 1 + (2.11 + 3.28i)T + (-7.06 + 15.4i)T^{2} \)
19 \( 1 + (-4.15 - 2.66i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-3.23 + 2.07i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (5.44 + 6.28i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-4.74 + 0.682i)T + (35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.536 + 3.73i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (0.999 + 0.866i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + 6.16iT - 47T^{2} \)
53 \( 1 + (5.78 - 2.64i)T + (34.7 - 40.0i)T^{2} \)
59 \( 1 + (-0.542 + 1.18i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (-8.65 - 9.98i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (4.59 - 15.6i)T + (-56.3 - 36.2i)T^{2} \)
71 \( 1 + (9.62 + 2.82i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-3.34 + 5.20i)T + (-30.3 - 66.4i)T^{2} \)
79 \( 1 + (2.95 - 6.47i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (11.8 - 1.69i)T + (79.6 - 23.3i)T^{2} \)
89 \( 1 + (-3.58 + 4.13i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-11.2 - 1.61i)T + (93.0 + 27.3i)T^{2} \)
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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

−11.96833196760558992292371513770, −11.32582284503230646325383987066, −10.09001847681153271283301298815, −9.607235155372955392191165554089, −8.685537258413761766838956108124, −7.43919280320763187298832281636, −6.00496782366285356268119163846, −4.24546921143739634323551555561, −3.51234697036857538834673685058, −2.49669603095195071277447339780, 1.58108631068604077480066121710, 3.49582183078577753797006774970, 4.69743645062703114462155879784, 6.34404924234570092839728752032, 7.22252098174310843039584520782, 8.132251774657941793467696499940, 8.960138185036438831533021109684, 9.618152384437183041529341953187, 11.65041600890076816611013764430, 12.55816877712930483627422195954

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