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Properties

Label 2-340-85.27-c1-0-4
Degree $2$
Conductor $340$
Sign $0.996 - 0.0870i$
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  + (1.49 − 0.296i)3-s + (2.12 − 0.684i)5-s + (2.36 + 3.53i)7-s + (−0.637 + 0.264i)9-s + (−1.78 − 2.67i)11-s + 4.88i·13-s + (2.96 − 1.65i)15-s + (−1.01 − 3.99i)17-s + (−4.80 − 1.98i)19-s + (4.56 + 4.56i)21-s + (0.757 − 3.80i)23-s + (4.06 − 2.91i)25-s + (−4.66 + 3.11i)27-s + (6.58 − 1.30i)29-s + (−2.09 + 3.14i)31-s + ⋯
L(s)  = 1  + (0.860 − 0.171i)3-s + (0.951 − 0.306i)5-s + (0.892 + 1.33i)7-s + (−0.212 + 0.0880i)9-s + (−0.539 − 0.807i)11-s + 1.35i·13-s + (0.766 − 0.426i)15-s + (−0.245 − 0.969i)17-s + (−1.10 − 0.456i)19-s + (0.997 + 0.997i)21-s + (0.157 − 0.793i)23-s + (0.812 − 0.583i)25-s + (−0.897 + 0.599i)27-s + (1.22 − 0.243i)29-s + (−0.377 + 0.564i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(340\)    =    \(2^{2} \cdot 5 \cdot 17\)
Sign: $0.996 - 0.0870i$
Analytic conductor: \(2.71491\)
Root analytic conductor: \(1.64769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{340} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 340,\ (\ :1/2),\ 0.996 - 0.0870i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97937 + 0.0862822i\)
\(L(\frac12)\) \(\approx\) \(1.97937 + 0.0862822i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.684i)T \)
17 \( 1 + (1.01 + 3.99i)T \)
good3 \( 1 + (-1.49 + 0.296i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (-2.36 - 3.53i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (1.78 + 2.67i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 - 4.88iT - 13T^{2} \)
19 \( 1 + (4.80 + 1.98i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.757 + 3.80i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-6.58 + 1.30i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + (2.09 - 3.14i)T + (-11.8 - 28.6i)T^{2} \)
37 \( 1 + (2.07 + 10.4i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (4.07 + 0.810i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (-6.03 - 2.50i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 7.93T + 47T^{2} \)
53 \( 1 + (-0.837 - 2.02i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.66 - 8.85i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-1.78 + 8.95i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (0.847 - 0.847i)T - 67iT^{2} \)
71 \( 1 + (4.58 + 3.06i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (-7.21 + 10.8i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (11.7 - 7.82i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (5.17 - 2.14i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-1.27 + 1.27i)T - 89iT^{2} \)
97 \( 1 + (4.70 - 7.04i)T + (-37.1 - 89.6i)T^{2} \)
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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

−11.54073778130853916332824844708, −10.69960058317034697462025237014, −9.166066185533105057149238385295, −8.881809361705523092498958311885, −8.167425123119686551602320025837, −6.66994464261784984270968086177, −5.58557719581865985699727222154, −4.68591167677668403404727746014, −2.67089642319196239849451723497, −2.06274330172902908649905071689, 1.73840393852557954609032750093, 3.08542270675791330930706086259, 4.35763082533502821609446635594, 5.56978128445250852878005652515, 6.84437354205474738842195823909, 7.959197719617205083326466151641, 8.504092296489742275913580815687, 10.04787845720064937281564830321, 10.24693113423823474604213301357, 11.23628921641662581189089832456

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