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Properties

Label 2-351-39.20-c1-0-4
Degree $2$
Conductor $351$
Sign $-0.101 - 0.994i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + (0.205 + 0.767i)2-s + (1.18 − 0.684i)4-s + (−2.75 + 2.75i)5-s + (1.54 + 0.414i)7-s + (1.89 + 1.89i)8-s + (−2.67 − 1.54i)10-s + (1.42 − 0.382i)11-s + (−1.55 + 3.25i)13-s + 1.27i·14-s + (0.306 − 0.530i)16-s + (−1.22 − 2.12i)17-s + (−2.03 + 7.59i)19-s + (−1.37 + 5.14i)20-s + (0.586 + 1.01i)22-s + (0.0635 − 0.110i)23-s + ⋯
L(s)  = 1  + (0.145 + 0.542i)2-s + (0.592 − 0.342i)4-s + (−1.23 + 1.23i)5-s + (0.583 + 0.156i)7-s + (0.669 + 0.669i)8-s + (−0.846 − 0.488i)10-s + (0.430 − 0.115i)11-s + (−0.432 + 0.901i)13-s + 0.339i·14-s + (0.0765 − 0.132i)16-s + (−0.297 − 0.515i)17-s + (−0.467 + 1.74i)19-s + (−0.308 + 1.15i)20-s + (0.125 + 0.216i)22-s + (0.0132 − 0.0229i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.101 - 0.994i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.101 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.965645 + 1.06966i\)
\(L(\frac12)\) \(\approx\) \(0.965645 + 1.06966i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (1.55 - 3.25i)T \)
good2 \( 1 + (-0.205 - 0.767i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (2.75 - 2.75i)T - 5iT^{2} \)
7 \( 1 + (-1.54 - 0.414i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.42 + 0.382i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.03 - 7.59i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.0635 + 0.110i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.00 - 4.04i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.87 - 1.87i)T + 31iT^{2} \)
37 \( 1 + (1.26 + 4.70i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.07 + 4.00i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.87 + 5.70i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.41 + 5.41i)T + 47iT^{2} \)
53 \( 1 + 7.39iT - 53T^{2} \)
59 \( 1 + (0.0877 - 0.327i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.120 + 0.208i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.41 - 1.45i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.902 + 0.241i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-4.50 + 4.50i)T - 73iT^{2} \)
79 \( 1 + 0.351T + 79T^{2} \)
83 \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \)
89 \( 1 + (-13.1 + 3.52i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.88 + 10.7i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.75006599230328939336792494927, −10.86852417429036946558801435159, −10.22735439736132547834568851135, −8.617414120750043738515361247151, −7.67734605092685782641126877588, −6.97707506634846119593935459004, −6.23150384827722976739005094650, −4.79844449895674261993548948274, −3.60794815743937185407122990427, −2.12129899559732631526433969356, 1.02682844034213722207822135330, 2.81447962831793619362727031785, 4.24977960018992519497952289454, 4.76931108833909468895294673923, 6.54687850191160369190930483863, 7.76651142346751784240666206319, 8.198182452868857032334867220000, 9.343746998240700448616251817189, 10.70166405574189044816321470951, 11.38964785130700872647753918510

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