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Properties

Label 2-430-215.79-c1-0-6
Degree $2$
Conductor $430$
Sign $-0.200 - 0.979i$
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 + (−0.377 + 0.217i)3-s − 4-s + (2.13 + 0.659i)5-s + (−0.217 − 0.377i)6-s + (0.236 + 0.136i)7-s − i·8-s + (−1.40 + 2.43i)9-s + (−0.659 + 2.13i)10-s + 3.88·11-s + (0.377 − 0.217i)12-s + (−0.559 − 0.323i)13-s + (−0.136 + 0.236i)14-s + (−0.950 + 0.216i)15-s + 16-s + (−1.73 − 1.00i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.217 + 0.125i)3-s − 0.5·4-s + (0.955 + 0.294i)5-s + (−0.0889 − 0.154i)6-s + (0.0893 + 0.0515i)7-s − 0.353i·8-s + (−0.468 + 0.811i)9-s + (−0.208 + 0.675i)10-s + 1.17·11-s + (0.108 − 0.0629i)12-s + (−0.155 − 0.0895i)13-s + (−0.0364 + 0.0631i)14-s + (−0.245 + 0.0559i)15-s + 0.250·16-s + (−0.421 − 0.243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.200 - 0.979i$
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: \((2,\ 430,\ (\ :1/2),\ -0.200 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895980 + 1.09760i\)
\(L(\frac12)\) \(\approx\) \(0.895980 + 1.09760i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.13 - 0.659i)T \)
43 \( 1 + (-4.54 + 4.73i)T \)
good3 \( 1 + (0.377 - 0.217i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.236 - 0.136i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.88T + 11T^{2} \)
13 \( 1 + (0.559 + 0.323i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.73 + 1.00i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.66 - 6.35i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.31 - 1.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.52 - 2.64i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.240 + 0.417i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.19 - 2.42i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.71T + 41T^{2} \)
47 \( 1 - 4.57iT - 47T^{2} \)
53 \( 1 + (-0.0419 + 0.0241i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.81T + 59T^{2} \)
61 \( 1 + (-3.10 + 5.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.80 + 5.08i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.77 + 9.99i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.41 + 3.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.35 - 2.35i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.28 + 5.36i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.0807 + 0.139i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.1iT - 97T^{2} \)
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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

−11.36692719388514894801743892940, −10.32708952828384473340443222485, −9.590090174156394489665215263916, −8.693528339077679731291262878384, −7.65075290841584439095674817350, −6.60710948003202699450585271291, −5.77039385996733172878516192628, −4.99426623645668977877263679767, −3.53928641196645145206320310469, −1.82955723567140878925942548960, 1.03074006123881617635361974628, 2.46404593205685922869641132037, 3.85186585552030271031319490411, 5.06477634162179957722440482294, 6.12007690324099542637264459103, 6.97035595067475843244112027510, 8.619136380645738144710146990352, 9.241486646646143367143461988438, 9.884290122264122579057776778411, 11.08992865426386868811737160502

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