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Properties

Label 2-340-17.4-c1-0-3
Degree $2$
Conductor $340$
Sign $-0.743 + 0.669i$
Analytic cond. $2.71491$
Root an. cond. $1.64769$
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  + (−1.16 + 1.16i)3-s + (−0.707 + 0.707i)5-s + (−2.24 − 2.24i)7-s + 0.275i·9-s + (−2.11 − 2.11i)11-s − 0.689·13-s − 1.65i·15-s + (−3.11 − 2.70i)17-s − 1.90i·19-s + 5.23·21-s + (−2.37 − 2.37i)23-s − 1.00i·25-s + (−3.82 − 3.82i)27-s + (−3.13 + 3.13i)29-s + (3.10 − 3.10i)31-s + ⋯
L(s)  = 1  + (−0.673 + 0.673i)3-s + (−0.316 + 0.316i)5-s + (−0.847 − 0.847i)7-s + 0.0917i·9-s + (−0.636 − 0.636i)11-s − 0.191·13-s − 0.426i·15-s + (−0.754 − 0.655i)17-s − 0.436i·19-s + 1.14·21-s + (−0.496 − 0.496i)23-s − 0.200i·25-s + (−0.735 − 0.735i)27-s + (−0.581 + 0.581i)29-s + (0.557 − 0.557i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(340\)    =    \(2^{2} \cdot 5 \cdot 17\)
Sign: $-0.743 + 0.669i$
Analytic conductor: \(2.71491\)
Root analytic conductor: \(1.64769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{340} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 340,\ (\ :1/2),\ -0.743 + 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0429421 - 0.111867i\)
\(L(\frac12)\) \(\approx\) \(0.0429421 - 0.111867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (3.11 + 2.70i)T \)
good3 \( 1 + (1.16 - 1.16i)T - 3iT^{2} \)
7 \( 1 + (2.24 + 2.24i)T + 7iT^{2} \)
11 \( 1 + (2.11 + 2.11i)T + 11iT^{2} \)
13 \( 1 + 0.689T + 13T^{2} \)
19 \( 1 + 1.90iT - 19T^{2} \)
23 \( 1 + (2.37 + 2.37i)T + 23iT^{2} \)
29 \( 1 + (3.13 - 3.13i)T - 29iT^{2} \)
31 \( 1 + (-3.10 + 3.10i)T - 31iT^{2} \)
37 \( 1 + (6.24 - 6.24i)T - 37iT^{2} \)
41 \( 1 + (-3.12 - 3.12i)T + 41iT^{2} \)
43 \( 1 - 2.71iT - 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + 0.432iT - 53T^{2} \)
59 \( 1 - 9.55iT - 59T^{2} \)
61 \( 1 + (-6.30 - 6.30i)T + 61iT^{2} \)
67 \( 1 - 8.05T + 67T^{2} \)
71 \( 1 + (6.69 - 6.69i)T - 71iT^{2} \)
73 \( 1 + (-6.43 + 6.43i)T - 73iT^{2} \)
79 \( 1 + (10.9 + 10.9i)T + 79iT^{2} \)
83 \( 1 - 8.14iT - 83T^{2} \)
89 \( 1 - 3.09T + 89T^{2} \)
97 \( 1 + (-1.31 + 1.31i)T - 97iT^{2} \)
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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

−11.04673760298224799861606187106, −10.36532403713697284778378251173, −9.662396257756138036699958720525, −8.339413411801878657706148874574, −7.22371971953694017881179669186, −6.31633828831550130809255909300, −5.10392670432579797629119931556, −4.12140787466137249200601669164, −2.86517627317646024205009282441, −0.085024200045742481498221151428, 2.05557093542852911382972420052, 3.69354963935130514350155046308, 5.21163769272579366207321594828, 6.11131538193263825395090933061, 6.96594101394087263450125158164, 8.047312045256527994767679167207, 9.134670862765522989088441393450, 10.00571645378496072350069165574, 11.18990799802745852290769747627, 12.10849205308356583686710722326

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