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Properties

Label 2-430-215.109-c1-0-13
Degree $2$
Conductor $430$
Sign $-0.547 + 0.836i$
Analytic cond. $3.43356$
Root an. cond. $1.85298$
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  + (−0.781 + 0.623i)2-s + (−1.57 + 0.618i)3-s + (0.222 − 0.974i)4-s + (0.731 − 2.11i)5-s + (0.847 − 1.46i)6-s + (0.588 − 0.339i)7-s + (0.433 + 0.900i)8-s + (−0.0950 + 0.0882i)9-s + (0.745 + 2.10i)10-s + (−0.0968 − 0.424i)11-s + (0.252 + 1.67i)12-s + (−1.14 + 1.67i)13-s + (−0.248 + 0.632i)14-s + (0.153 + 3.78i)15-s + (−0.900 − 0.433i)16-s + (−7.86 + 0.589i)17-s + ⋯
L(s)  = 1  + (−0.552 + 0.440i)2-s + (−0.910 + 0.357i)3-s + (0.111 − 0.487i)4-s + (0.327 − 0.944i)5-s + (0.345 − 0.598i)6-s + (0.222 − 0.128i)7-s + (0.153 + 0.318i)8-s + (−0.0316 + 0.0294i)9-s + (0.235 + 0.666i)10-s + (−0.0291 − 0.127i)11-s + (0.0728 + 0.483i)12-s + (−0.317 + 0.465i)13-s + (−0.0663 + 0.169i)14-s + (0.0397 + 0.977i)15-s + (−0.225 − 0.108i)16-s + (−1.90 + 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.547 + 0.836i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.547 + 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121017 - 0.223840i\)
\(L(\frac12)\) \(\approx\) \(0.121017 - 0.223840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 - 0.623i)T \)
5 \( 1 + (-0.731 + 2.11i)T \)
43 \( 1 + (6.54 - 0.333i)T \)
good3 \( 1 + (1.57 - 0.618i)T + (2.19 - 2.04i)T^{2} \)
7 \( 1 + (-0.588 + 0.339i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.0968 + 0.424i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (1.14 - 1.67i)T + (-4.74 - 12.1i)T^{2} \)
17 \( 1 + (7.86 - 0.589i)T + (16.8 - 2.53i)T^{2} \)
19 \( 1 + (3.41 + 3.16i)T + (1.41 + 18.9i)T^{2} \)
23 \( 1 + (-1.60 + 5.20i)T + (-19.0 - 12.9i)T^{2} \)
29 \( 1 + (2.46 - 6.27i)T + (-21.2 - 19.7i)T^{2} \)
31 \( 1 + (-9.17 + 1.38i)T + (29.6 - 9.13i)T^{2} \)
37 \( 1 + (4.70 + 2.71i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.14 + 7.69i)T + (-9.12 + 39.9i)T^{2} \)
47 \( 1 + (-1.46 - 0.333i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (6.49 + 9.51i)T + (-19.3 + 49.3i)T^{2} \)
59 \( 1 + (9.43 + 4.54i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-7.91 - 1.19i)T + (58.2 + 17.9i)T^{2} \)
67 \( 1 + (1.27 - 1.37i)T + (-5.00 - 66.8i)T^{2} \)
71 \( 1 + (6.28 - 1.93i)T + (58.6 - 39.9i)T^{2} \)
73 \( 1 + (4.41 - 6.47i)T + (-26.6 - 67.9i)T^{2} \)
79 \( 1 + (-4.76 - 8.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.55 + 1.00i)T + (60.8 - 56.4i)T^{2} \)
89 \( 1 + (2.46 + 6.27i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-8.18 + 1.86i)T + (87.3 - 42.0i)T^{2} \)
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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

−10.83666726556761630442091614821, −9.985971627187153976716426398757, −8.783102126445984462380711783889, −8.511156444480975652955942047762, −6.89794862351896830906342901656, −6.20609280924808218226848564615, −4.94367340738334421290730078937, −4.56919172297114015296414429702, −2.09513476866390965771835997326, −0.20333336919799282481533351231, 1.87819842243838996742730938068, 3.15465977811252579003148037224, 4.74607833178080113388977674528, 6.12837949740501261190252159168, 6.67548643820823361664813841785, 7.75752296501694842856442839281, 8.837311543106666949326733030293, 9.944019978017363723309865057613, 10.64308734293857089103036595943, 11.48066396391677815032688047743

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