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Properties

Label 2-430-215.79-c1-0-15
Degree $2$
Conductor $430$
Sign $0.562 + 0.826i$
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 + (2.19 − 1.26i)3-s − 4-s + (1.41 + 1.73i)5-s + (−1.26 − 2.19i)6-s + (2.46 + 1.42i)7-s + i·8-s + (1.71 − 2.96i)9-s + (1.73 − 1.41i)10-s − 2.30·11-s + (−2.19 + 1.26i)12-s + (3.57 + 2.06i)13-s + (1.42 − 2.46i)14-s + (5.29 + 2.01i)15-s + 16-s + (−5.72 − 3.30i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (1.26 − 0.731i)3-s − 0.5·4-s + (0.631 + 0.775i)5-s + (−0.517 − 0.896i)6-s + (0.931 + 0.537i)7-s + 0.353i·8-s + (0.571 − 0.989i)9-s + (0.548 − 0.446i)10-s − 0.693·11-s + (−0.633 + 0.365i)12-s + (0.990 + 0.571i)13-s + (0.380 − 0.658i)14-s + (1.36 + 0.520i)15-s + 0.250·16-s + (−1.38 − 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.562 + 0.826i$
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: \((2,\ 430,\ (\ :1/2),\ 0.562 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97921 - 1.04661i\)
\(L(\frac12)\) \(\approx\) \(1.97921 - 1.04661i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - 1.73i)T \)
43 \( 1 + (1.42 + 6.40i)T \)
good3 \( 1 + (-2.19 + 1.26i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.46 - 1.42i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.30T + 11T^{2} \)
13 \( 1 + (-3.57 - 2.06i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.72 + 3.30i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.326 + 0.565i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.10 - 0.636i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.34 + 4.05i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.88 + 8.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.89 - 2.82i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
47 \( 1 - 2.33iT - 47T^{2} \)
53 \( 1 + (-4.83 + 2.79i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.86T + 59T^{2} \)
61 \( 1 + (4.28 - 7.42i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.4 + 6.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.25 - 3.91i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.03 - 5.21i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.70 + 2.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.60 + 4.96i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.89 + 5.01i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.24iT - 97T^{2} \)
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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

−11.16675783766405998741901885105, −10.10392412268316377375553067855, −9.020939055080444212722629828098, −8.515738713693397971182536573519, −7.52070111688368280465688259131, −6.48212216477706541036177719275, −5.12505058836335138528843384978, −3.63215604691245221982334320442, −2.36292490474575622497221888142, −1.94752914910884766245089307226, 1.78554053649927854486181394097, 3.49790939200382998535726361249, 4.55475730113007526318811812128, 5.33339156338272491265511739554, 6.70262453368280314502855235192, 8.160335869255523579038926015415, 8.423771389956262333652580996437, 9.136431392988306883926004618265, 10.30453017611862502850345501954, 10.81941806310435245325733219774

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