Properties

Label 2-230-115.59-c1-0-4
Degree $2$
Conductor $230$
Sign $0.894 + 0.446i$
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  + (−0.755 + 0.654i)2-s + (−1.37 + 2.13i)3-s + (0.142 − 0.989i)4-s + (−0.179 − 2.22i)5-s + (−0.360 − 2.50i)6-s + (1.23 − 4.20i)7-s + (0.540 + 0.841i)8-s + (−1.42 − 3.12i)9-s + (1.59 + 1.56i)10-s + (−1.72 + 1.99i)11-s + (1.91 + 1.66i)12-s + (−1.45 − 4.94i)13-s + (1.82 + 3.98i)14-s + (5.00 + 2.67i)15-s + (−0.959 − 0.281i)16-s + (5.70 − 0.819i)17-s + ⋯
L(s)  = 1  + (−0.534 + 0.463i)2-s + (−0.791 + 1.23i)3-s + (0.0711 − 0.494i)4-s + (−0.0801 − 0.996i)5-s + (−0.147 − 1.02i)6-s + (0.466 − 1.58i)7-s + (0.191 + 0.297i)8-s + (−0.474 − 1.04i)9-s + (0.504 + 0.495i)10-s + (−0.520 + 0.600i)11-s + (0.553 + 0.479i)12-s + (−0.402 − 1.37i)13-s + (0.486 + 1.06i)14-s + (1.29 + 0.690i)15-s + (−0.239 − 0.0704i)16-s + (1.38 − 0.198i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.894 + 0.446i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.894 + 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.626500 - 0.147670i\)
\(L(\frac12)\) \(\approx\) \(0.626500 - 0.147670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.755 - 0.654i)T \)
5 \( 1 + (0.179 + 2.22i)T \)
23 \( 1 + (-3.67 - 3.08i)T \)
good3 \( 1 + (1.37 - 2.13i)T + (-1.24 - 2.72i)T^{2} \)
7 \( 1 + (-1.23 + 4.20i)T + (-5.88 - 3.78i)T^{2} \)
11 \( 1 + (1.72 - 1.99i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (1.45 + 4.94i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-5.70 + 0.819i)T + (16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.0206 + 0.143i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (1.25 + 8.71i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-1.81 + 1.16i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (6.49 - 2.96i)T + (24.2 - 27.9i)T^{2} \)
41 \( 1 + (-1.36 + 2.97i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-5.42 + 8.44i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 - 3.07iT - 47T^{2} \)
53 \( 1 + (0.336 - 1.14i)T + (-44.5 - 28.6i)T^{2} \)
59 \( 1 + (5.12 - 1.50i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (-4.96 + 3.18i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (0.813 - 0.704i)T + (9.53 - 66.3i)T^{2} \)
71 \( 1 + (-8.44 - 9.74i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-5.76 - 0.828i)T + (70.0 + 20.5i)T^{2} \)
79 \( 1 + (3.12 - 0.916i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (4.31 - 1.96i)T + (54.3 - 62.7i)T^{2} \)
89 \( 1 + (-4.05 - 2.60i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (7.29 + 3.33i)T + (63.5 + 73.3i)T^{2} \)
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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

−11.89626305302938114613973686691, −10.77526156773178486027303802180, −10.14316813691351163999314831816, −9.579612681702667104226135488917, −8.023943869764343660084451829851, −7.42041001775010879989291846863, −5.54879455372636230739494715133, −4.98445889930332768231368411107, −3.91590700321520730865646521075, −0.72297992063678373717896853131, 1.77896791592366184961017332953, 2.96063184762259701357609174505, 5.30973631515818804015160443739, 6.33534232744970360064688139326, 7.25108038669403288237183303885, 8.245483255735211459977781919453, 9.323366285638989397475158975892, 10.72676325680171903237890876138, 11.43873750121888797699964474454, 12.12702679183774458425674189521

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