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Properties

Label 2-357-119.16-c1-0-11
Degree $2$
Conductor $357$
Sign $-0.202 - 0.979i$
Analytic cond. $2.85065$
Root an. cond. $1.68838$
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  + (−0.427 + 0.740i)2-s + (−0.866 + 0.5i)3-s + (0.634 + 1.09i)4-s + (3.01 + 1.73i)5-s − 0.855i·6-s + (2.30 + 1.29i)7-s − 2.79·8-s + (0.499 − 0.866i)9-s + (−2.57 + 1.48i)10-s + (4.73 − 2.73i)11-s + (−1.09 − 0.634i)12-s − 1.39·13-s + (−1.94 + 1.15i)14-s − 3.47·15-s + (−0.0728 + 0.126i)16-s + (1.58 − 3.80i)17-s + ⋯
L(s)  = 1  + (−0.302 + 0.523i)2-s + (−0.499 + 0.288i)3-s + (0.317 + 0.549i)4-s + (1.34 + 0.778i)5-s − 0.349i·6-s + (0.872 + 0.489i)7-s − 0.988·8-s + (0.166 − 0.288i)9-s + (−0.815 + 0.470i)10-s + (1.42 − 0.823i)11-s + (−0.317 − 0.183i)12-s − 0.385·13-s + (−0.520 + 0.308i)14-s − 0.898·15-s + (−0.0182 + 0.0315i)16-s + (0.385 − 0.922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $-0.202 - 0.979i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ -0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904087 + 1.10999i\)
\(L(\frac12)\) \(\approx\) \(0.904087 + 1.10999i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-2.30 - 1.29i)T \)
17 \( 1 + (-1.58 + 3.80i)T \)
good2 \( 1 + (0.427 - 0.740i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-3.01 - 1.73i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.73 + 2.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.39T + 13T^{2} \)
19 \( 1 + (0.0543 - 0.0941i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.77 + 3.91i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.259iT - 29T^{2} \)
31 \( 1 + (4.32 - 2.49i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.259 - 0.149i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.70iT - 41T^{2} \)
43 \( 1 + 9.73T + 43T^{2} \)
47 \( 1 + (-2.55 + 4.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.04 + 8.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.27 + 5.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.39 + 1.38i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.35 + 2.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.8iT - 71T^{2} \)
73 \( 1 + (-10.2 + 5.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.707 - 0.408i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.882T + 83T^{2} \)
89 \( 1 + (6.35 - 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.7iT - 97T^{2} \)
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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

−11.67791897367527144065460222531, −10.85907533182427198252973296325, −9.727232251569540559202560657471, −9.010530361731322281480566557484, −7.944497919056264401278379500246, −6.63965557553047803763951589600, −6.21361139364424659627964821286, −5.13845211289649423344830470005, −3.42314595129895606992151634326, −2.04095014972311167864213267717, 1.44406824057891381732223577079, 1.85050767404543473518230400056, 4.26316189355951175120031562570, 5.49284050534136725047423012218, 6.14462498599119817351440830640, 7.28509996192239093079176825694, 8.693718952068696141514003045052, 9.706084325484186760201266490107, 10.10491833966465494613650839676, 11.20302676827946040892862701972

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