Properties

Label 2-342-171.59-c1-0-17
Degree $2$
Conductor $342$
Sign $0.473 + 0.880i$
Analytic cond. $2.73088$
Root an. cond. $1.65253$
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  + (0.939 − 0.342i)2-s + (0.441 − 1.67i)3-s + (0.766 − 0.642i)4-s + (2.80 + 0.493i)5-s + (−0.158 − 1.72i)6-s + (0.343 − 0.595i)7-s + (0.500 − 0.866i)8-s + (−2.61 − 1.47i)9-s + (2.80 − 0.493i)10-s + 4.77i·11-s + (−0.738 − 1.56i)12-s + (−1.94 + 0.342i)13-s + (0.119 − 0.677i)14-s + (2.06 − 4.47i)15-s + (0.173 − 0.984i)16-s + (−1.42 − 0.251i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.254 − 0.967i)3-s + (0.383 − 0.321i)4-s + (1.25 + 0.220i)5-s + (−0.0646 − 0.704i)6-s + (0.130 − 0.225i)7-s + (0.176 − 0.306i)8-s + (−0.870 − 0.492i)9-s + (0.885 − 0.156i)10-s + 1.44i·11-s + (−0.213 − 0.452i)12-s + (−0.538 + 0.0948i)13-s + (0.0319 − 0.181i)14-s + (0.532 − 1.15i)15-s + (0.0434 − 0.246i)16-s + (−0.345 − 0.0610i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.473 + 0.880i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.473 + 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02262 - 1.20911i\)
\(L(\frac12)\) \(\approx\) \(2.02262 - 1.20911i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (-0.441 + 1.67i)T \)
19 \( 1 + (1.55 - 4.07i)T \)
good5 \( 1 + (-2.80 - 0.493i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.343 + 0.595i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.77iT - 11T^{2} \)
13 \( 1 + (1.94 - 0.342i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.42 + 0.251i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (3.50 + 4.17i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.60 + 2.18i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + 4.18iT - 31T^{2} \)
37 \( 1 - 9.14iT - 37T^{2} \)
41 \( 1 + (-4.53 + 1.65i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (8.90 + 7.47i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (6.84 + 8.15i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-8.96 - 3.26i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-5.84 - 4.90i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.47 - 8.34i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.91 - 8.01i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-5.24 + 1.90i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-2.37 - 1.99i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-2.18 - 0.386i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (8.42 + 4.86i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.30 + 5.28i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (2.69 + 7.39i)T + (-74.3 + 62.3i)T^{2} \)
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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

−11.77236468043137149779610069608, −10.25268722855151332576858048024, −9.863220354431420100307089835566, −8.474657584161253498352286453811, −7.24661719089861852640002319712, −6.51056862421958483673971003581, −5.59107366029360538012521805426, −4.28895945467187891577555293728, −2.49988850988958903086154631630, −1.81459455355844035208841664532, 2.33053698502517367926619374228, 3.49182484962223409424991491133, 4.90628209901238390611428075031, 5.58310123026900697043794754586, 6.47717847257285629147788366128, 8.108552227816650156068178277061, 8.988366269200523117762618643978, 9.776476086486381851289606823240, 10.81942716211330044809829152318, 11.51848392485598413303244959366

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