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Properties

Label 2-340-85.57-c1-0-4
Degree $2$
Conductor $340$
Sign $0.996 + 0.0847i$
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  + (0.286 − 0.428i)3-s + (1.53 + 1.62i)5-s + (0.632 − 3.17i)7-s + (1.04 + 2.52i)9-s + (0.276 − 1.38i)11-s + 1.82i·13-s + (1.13 − 0.192i)15-s + (3.00 − 2.82i)17-s + (−0.686 + 1.65i)19-s + (−1.17 − 1.17i)21-s + (2.79 − 1.86i)23-s + (−0.282 + 4.99i)25-s + (2.89 + 0.576i)27-s + (0.0112 − 0.0168i)29-s + (0.194 + 0.978i)31-s + ⋯
L(s)  = 1  + (0.165 − 0.247i)3-s + (0.686 + 0.726i)5-s + (0.238 − 1.20i)7-s + (0.348 + 0.842i)9-s + (0.0832 − 0.418i)11-s + 0.507i·13-s + (0.293 − 0.0497i)15-s + (0.728 − 0.684i)17-s + (−0.157 + 0.380i)19-s + (−0.257 − 0.257i)21-s + (0.582 − 0.389i)23-s + (−0.0564 + 0.998i)25-s + (0.557 + 0.110i)27-s + (0.00209 − 0.00313i)29-s + (0.0349 + 0.175i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(340\)    =    \(2^{2} \cdot 5 \cdot 17\)
Sign: $0.996 + 0.0847i$
Analytic conductor: \(2.71491\)
Root analytic conductor: \(1.64769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{340} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 340,\ (\ :1/2),\ 0.996 + 0.0847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63242 - 0.0693206i\)
\(L(\frac12)\) \(\approx\) \(1.63242 - 0.0693206i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.53 - 1.62i)T \)
17 \( 1 + (-3.00 + 2.82i)T \)
good3 \( 1 + (-0.286 + 0.428i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (-0.632 + 3.17i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (-0.276 + 1.38i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 - 1.82iT - 13T^{2} \)
19 \( 1 + (0.686 - 1.65i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.79 + 1.86i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-0.0112 + 0.0168i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-0.194 - 0.978i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (4.40 + 2.94i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (4.69 + 7.02i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.715 - 1.72i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 0.949T + 47T^{2} \)
53 \( 1 + (2.97 - 1.23i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (13.1 - 5.46i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.52 - 2.35i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (2.65 - 2.65i)T - 67iT^{2} \)
71 \( 1 + (4.45 - 0.885i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-2.66 - 13.4i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (13.2 + 2.64i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (5.03 + 12.1i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (8.46 - 8.46i)T - 89iT^{2} \)
97 \( 1 + (1.78 + 8.95i)T + (-89.6 + 37.1i)T^{2} \)
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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

−11.26375989986163395557398541304, −10.55769872404123624565107644200, −9.933150340932312734350970304577, −8.686247265219959392612741820008, −7.42536942918976205524973795234, −7.00749527859045421494477693921, −5.68950254643247103422238189074, −4.44052176760403564826652917175, −3.06267677115528192213409362640, −1.57860330369495990826523007902, 1.59772275074854041701358510972, 3.13496553839407974576914952732, 4.67110495981049778167113519069, 5.59414536334380031693381888210, 6.51931849075722979730536444246, 8.026073477413577579532370694133, 8.928216049600062313812171595652, 9.533653903712182095966720454279, 10.41024815537168674478373893968, 11.79620861960041365606579597719

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