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Properties

Label 2-3460-3460.443-c0-0-0
Degree $2$
Conductor $3460$
Sign $-0.802 + 0.596i$
Analytic cond. $1.72676$
Root an. cond. $1.31406$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.145 + 0.989i)2-s + (−0.957 + 0.288i)4-s + (−0.217 + 0.976i)5-s + (−0.424 − 0.905i)8-s + (0.0365 + 0.999i)9-s + (−0.997 − 0.0729i)10-s + (−1.15 + 1.56i)13-s + (0.833 − 0.551i)16-s + (0.803 + 0.274i)17-s + (−0.983 + 0.181i)18-s + (−0.0729 − 0.997i)20-s + (−0.905 − 0.424i)25-s + (−1.71 − 0.920i)26-s + (0.411 − 0.294i)29-s + (0.667 + 0.744i)32-s + ⋯
L(s)  = 1  + (0.145 + 0.989i)2-s + (−0.957 + 0.288i)4-s + (−0.217 + 0.976i)5-s + (−0.424 − 0.905i)8-s + (0.0365 + 0.999i)9-s + (−0.997 − 0.0729i)10-s + (−1.15 + 1.56i)13-s + (0.833 − 0.551i)16-s + (0.803 + 0.274i)17-s + (−0.983 + 0.181i)18-s + (−0.0729 − 0.997i)20-s + (−0.905 − 0.424i)25-s + (−1.71 − 0.920i)26-s + (0.411 − 0.294i)29-s + (0.667 + 0.744i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3460\)    =    \(2^{2} \cdot 5 \cdot 173\)
Sign: $-0.802 + 0.596i$
Analytic conductor: \(1.72676\)
Root analytic conductor: \(1.31406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3460} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3460,\ (\ :0),\ -0.802 + 0.596i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8065633203\)
\(L(\frac12)\) \(\approx\) \(0.8065633203\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.145 - 0.989i)T \)
5 \( 1 + (0.217 - 0.976i)T \)
173 \( 1 + (0.551 + 0.833i)T \)
good3 \( 1 + (-0.0365 - 0.999i)T^{2} \)
7 \( 1 + (0.872 + 0.489i)T^{2} \)
11 \( 1 + (-0.424 - 0.905i)T^{2} \)
13 \( 1 + (1.15 - 1.56i)T + (-0.288 - 0.957i)T^{2} \)
17 \( 1 + (-0.803 - 0.274i)T + (0.791 + 0.611i)T^{2} \)
19 \( 1 + (0.217 + 0.976i)T^{2} \)
23 \( 1 + (0.424 - 0.905i)T^{2} \)
29 \( 1 + (-0.411 + 0.294i)T + (0.322 - 0.946i)T^{2} \)
31 \( 1 + (-0.0365 + 0.999i)T^{2} \)
37 \( 1 + (-0.468 + 0.583i)T + (-0.217 - 0.976i)T^{2} \)
41 \( 1 + (1.88 + 0.493i)T + (0.872 + 0.489i)T^{2} \)
43 \( 1 + (0.551 - 0.833i)T^{2} \)
47 \( 1 + (-0.145 + 0.989i)T^{2} \)
53 \( 1 + (-1.15 - 1.49i)T + (-0.252 + 0.967i)T^{2} \)
59 \( 1 + (-0.853 + 0.520i)T^{2} \)
61 \( 1 + (-0.163 - 0.0804i)T + (0.611 + 0.791i)T^{2} \)
67 \( 1 + (-0.999 + 0.0365i)T^{2} \)
71 \( 1 + (0.946 + 0.322i)T^{2} \)
73 \( 1 + (1.18 + 0.108i)T + (0.983 + 0.181i)T^{2} \)
79 \( 1 + (0.145 + 0.989i)T^{2} \)
83 \( 1 + (-0.719 + 0.694i)T^{2} \)
89 \( 1 + (1.24 + 1.11i)T + (0.109 + 0.994i)T^{2} \)
97 \( 1 + (0.900 - 0.382i)T + (0.694 - 0.719i)T^{2} \)
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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

−9.037916648885089239452710359951, −8.273005729380048453030372563339, −7.44006510637903316226530354367, −7.16076090350637920199306935509, −6.36879655650238504724283287665, −5.53163485200422398130581252426, −4.69153331864463941881227920271, −4.03644657315511125622452349292, −2.98833742259444803805740437785, −1.95517651162445904345102048355, 0.46538817728294350859805287447, 1.45761947189725926705199134215, 2.84875138094804891063215367904, 3.43435289038969077725377218379, 4.42478466341387203666151770173, 5.19573517322206603981265298372, 5.64596082967240725260847194171, 6.84978738264682312137579531297, 7.956524067610270973620694129506, 8.358018104269289784848239357163

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