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Properties

Label 2-340-17.2-c1-0-5
Degree $2$
Conductor $340$
Sign $0.649 + 0.760i$
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  + (2.48 − 1.02i)3-s + (−0.382 − 0.923i)5-s + (0.413 − 0.999i)7-s + (2.99 − 2.99i)9-s + (−1.06 − 0.439i)11-s + 2.13i·13-s + (−1.90 − 1.90i)15-s + (−1.45 − 3.85i)17-s + (3.89 + 3.89i)19-s − 2.90i·21-s + (1.15 + 0.479i)23-s + (−0.707 + 0.707i)25-s + (1.26 − 3.05i)27-s + (1.78 + 4.31i)29-s + (−8.95 + 3.71i)31-s + ⋯
L(s)  = 1  + (1.43 − 0.594i)3-s + (−0.171 − 0.413i)5-s + (0.156 − 0.377i)7-s + (0.996 − 0.996i)9-s + (−0.320 − 0.132i)11-s + 0.591i·13-s + (−0.490 − 0.490i)15-s + (−0.352 − 0.935i)17-s + (0.894 + 0.894i)19-s − 0.634i·21-s + (0.241 + 0.0999i)23-s + (−0.141 + 0.141i)25-s + (0.243 − 0.587i)27-s + (0.331 + 0.801i)29-s + (−1.60 + 0.666i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(340\)    =    \(2^{2} \cdot 5 \cdot 17\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(2.71491\)
Root analytic conductor: \(1.64769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{340} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 340,\ (\ :1/2),\ 0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81239 - 0.835616i\)
\(L(\frac12)\) \(\approx\) \(1.81239 - 0.835616i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.382 + 0.923i)T \)
17 \( 1 + (1.45 + 3.85i)T \)
good3 \( 1 + (-2.48 + 1.02i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (-0.413 + 0.999i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.06 + 0.439i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 2.13iT - 13T^{2} \)
19 \( 1 + (-3.89 - 3.89i)T + 19iT^{2} \)
23 \( 1 + (-1.15 - 0.479i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.78 - 4.31i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (8.95 - 3.71i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (2.07 - 0.859i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.51 + 3.65i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-4.05 + 4.05i)T - 43iT^{2} \)
47 \( 1 - 1.02iT - 47T^{2} \)
53 \( 1 + (-4.13 - 4.13i)T + 53iT^{2} \)
59 \( 1 + (-2.45 + 2.45i)T - 59iT^{2} \)
61 \( 1 + (0.102 - 0.246i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 2.16T + 67T^{2} \)
71 \( 1 + (11.9 - 4.94i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-5.26 - 12.7i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (15.1 + 6.28i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (12.1 + 12.1i)T + 83iT^{2} \)
89 \( 1 + 10.4iT - 89T^{2} \)
97 \( 1 + (-4.97 - 12.0i)T + (-68.5 + 68.5i)T^{2} \)
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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

−11.55421966065536096612633498535, −10.33848865012727950585142353145, −9.152234245312919406470381057601, −8.719479547155220796607385024534, −7.52657352241670823673002254167, −7.14102002339306631337127646329, −5.41723075547730704540823694544, −4.02755333438495369413992352200, −2.91941394936032945634068014324, −1.52086993069626474068238352655, 2.30094280323651390857364916292, 3.26108948477202750472349698720, 4.33777704708319002686881682104, 5.68040980137156257561204345747, 7.21160280164819581904738343138, 8.053986153435447587307179454242, 8.861927495194868318637683224196, 9.682867824923073713488296864030, 10.56285476233580666061999842323, 11.49997038913444222882351481427

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