Properties

Label 2-430-5.4-c1-0-7
Degree $2$
Conductor $430$
Sign $-0.792 - 0.609i$
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 + 1.20i·3-s − 4-s + (1.77 + 1.36i)5-s − 1.20·6-s + 4.33i·7-s i·8-s + 1.55·9-s + (−1.36 + 1.77i)10-s + 0.276·11-s − 1.20i·12-s − 4.16i·13-s − 4.33·14-s + (−1.63 + 2.12i)15-s + 16-s − 2.65i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.692i·3-s − 0.5·4-s + (0.792 + 0.609i)5-s − 0.489·6-s + 1.63i·7-s − 0.353i·8-s + 0.519·9-s + (−0.431 + 0.560i)10-s + 0.0834·11-s − 0.346i·12-s − 1.15i·13-s − 1.15·14-s + (−0.422 + 0.549i)15-s + 0.250·16-s − 0.644i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.792 - 0.609i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.792 - 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487136 + 1.43235i\)
\(L(\frac12)\) \(\approx\) \(0.487136 + 1.43235i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.77 - 1.36i)T \)
43 \( 1 + iT \)
good3 \( 1 - 1.20iT - 3T^{2} \)
7 \( 1 - 4.33iT - 7T^{2} \)
11 \( 1 - 0.276T + 11T^{2} \)
13 \( 1 + 4.16iT - 13T^{2} \)
17 \( 1 + 2.65iT - 17T^{2} \)
19 \( 1 + 1.21T + 19T^{2} \)
23 \( 1 - 1.82iT - 23T^{2} \)
29 \( 1 + 8.13T + 29T^{2} \)
31 \( 1 - 0.868T + 31T^{2} \)
37 \( 1 + 5.72iT - 37T^{2} \)
41 \( 1 - 4.91T + 41T^{2} \)
47 \( 1 + 7.94iT - 47T^{2} \)
53 \( 1 + 1.31iT - 53T^{2} \)
59 \( 1 - 4.68T + 59T^{2} \)
61 \( 1 - 5.55T + 61T^{2} \)
67 \( 1 + 7.06iT - 67T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 - 4.44iT - 73T^{2} \)
79 \( 1 + 3.37T + 79T^{2} \)
83 \( 1 - 9.61iT - 83T^{2} \)
89 \( 1 - 8.27T + 89T^{2} \)
97 \( 1 - 14.9iT - 97T^{2} \)
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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

−11.36656316538805131157576193962, −10.38993610123724291912813325242, −9.510965388959224269589739624350, −9.048647454327489548402657237015, −7.82825129154186761377041679154, −6.73871932919303431279315854434, −5.59416710365025301451699780060, −5.26109418355224458079036626883, −3.60263096606264280688480810279, −2.30198201544671123352662708969, 1.07592533165659049076516236618, 1.99486796780707926286156248128, 3.93075861511654830080920282205, 4.63158447307650221561587946019, 6.20145653083436478877982539120, 7.07175192787216514154352989073, 8.054505612090371179269248080834, 9.226051945619262478794314780499, 9.992354396168651780008480658891, 10.73725417968086438108137927972

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