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Properties

Label 2-430-215.79-c1-0-14
Degree $2$
Conductor $430$
Sign $-0.196 + 0.980i$
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  − i·2-s + (−1.66 + 0.959i)3-s − 4-s + (1.57 − 1.58i)5-s + (0.959 + 1.66i)6-s + (−0.863 − 0.498i)7-s + i·8-s + (0.340 − 0.590i)9-s + (−1.58 − 1.57i)10-s + 6.30·11-s + (1.66 − 0.959i)12-s + (−2.37 − 1.37i)13-s + (−0.498 + 0.863i)14-s + (−1.10 + 4.14i)15-s + 16-s + (−5.74 − 3.31i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.959 + 0.553i)3-s − 0.5·4-s + (0.705 − 0.708i)5-s + (0.391 + 0.678i)6-s + (−0.326 − 0.188i)7-s + 0.353i·8-s + (0.113 − 0.196i)9-s + (−0.501 − 0.498i)10-s + 1.90·11-s + (0.479 − 0.276i)12-s + (−0.659 − 0.380i)13-s + (−0.133 + 0.230i)14-s + (−0.284 + 1.07i)15-s + 0.250·16-s + (−1.39 − 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.196 + 0.980i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.196 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605730 - 0.739551i\)
\(L(\frac12)\) \(\approx\) \(0.605730 - 0.739551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-1.57 + 1.58i)T \)
43 \( 1 + (-5.09 - 4.12i)T \)
good3 \( 1 + (1.66 - 0.959i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.863 + 0.498i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 6.30T + 11T^{2} \)
13 \( 1 + (2.37 + 1.37i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.74 + 3.31i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.14 + 1.97i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.44 + 4.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.00 + 5.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.10 + 3.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.618 + 0.357i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.42T + 41T^{2} \)
47 \( 1 - 5.64iT - 47T^{2} \)
53 \( 1 + (1.14 - 0.659i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.10T + 59T^{2} \)
61 \( 1 + (6.35 - 11.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.02 + 2.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.60 - 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.16 - 4.71i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.53 + 9.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.92 - 5.72i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.39 - 2.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.30iT - 97T^{2} \)
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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

−11.07021302834255725233188635063, −10.00704478872607030450563131243, −9.343732890904463076064409646429, −8.680496310385566695494557522773, −6.86793083569894378192854464387, −6.01829093246052671500329839722, −4.79200322822102322225815166673, −4.32206492731924100530890788479, −2.49848458754999737658587724485, −0.73572950115476556664169693565, 1.59696146935184333435412778500, 3.54519918784549455094958588080, 5.00441539435585736398294956714, 6.14065342420853857502966382324, 6.65224190137230979363480427302, 7.11333311482303424182191700646, 8.921282311081782323171253206211, 9.334991798294838521096440606755, 10.64739138445877804645036324134, 11.41703945958034070692564028872

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