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Properties

Label 2-354-177.101-c1-0-13
Degree $2$
Conductor $354$
Sign $0.969 + 0.245i$
Analytic cond. $2.82670$
Root an. cond. $1.68128$
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  + (−0.561 + 0.827i)2-s + (1.51 + 0.835i)3-s + (−0.370 − 0.928i)4-s + (1.02 − 2.20i)5-s + (−1.54 + 0.787i)6-s + (−1.16 − 4.21i)7-s + (0.976 + 0.214i)8-s + (1.60 + 2.53i)9-s + (1.25 + 2.08i)10-s + (−0.616 − 0.584i)11-s + (0.214 − 1.71i)12-s + (0.461 − 4.24i)13-s + (4.14 + 1.39i)14-s + (3.39 − 2.49i)15-s + (−0.725 + 0.687i)16-s + (1.97 + 0.547i)17-s + ⋯
L(s)  = 1  + (−0.396 + 0.585i)2-s + (0.876 + 0.482i)3-s + (−0.185 − 0.464i)4-s + (0.456 − 0.987i)5-s + (−0.629 + 0.321i)6-s + (−0.441 − 1.59i)7-s + (0.345 + 0.0760i)8-s + (0.534 + 0.844i)9-s + (0.396 + 0.659i)10-s + (−0.186 − 0.176i)11-s + (0.0618 − 0.496i)12-s + (0.128 − 1.17i)13-s + (1.10 + 0.372i)14-s + (0.876 − 0.644i)15-s + (−0.181 + 0.171i)16-s + (0.478 + 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.969 + 0.245i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ 0.969 + 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43352 - 0.178892i\)
\(L(\frac12)\) \(\approx\) \(1.43352 - 0.178892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.561 - 0.827i)T \)
3 \( 1 + (-1.51 - 0.835i)T \)
59 \( 1 + (7.12 - 2.86i)T \)
good5 \( 1 + (-1.02 + 2.20i)T + (-3.23 - 3.81i)T^{2} \)
7 \( 1 + (1.16 + 4.21i)T + (-5.99 + 3.60i)T^{2} \)
11 \( 1 + (0.616 + 0.584i)T + (0.595 + 10.9i)T^{2} \)
13 \( 1 + (-0.461 + 4.24i)T + (-12.6 - 2.79i)T^{2} \)
17 \( 1 + (-1.97 - 0.547i)T + (14.5 + 8.76i)T^{2} \)
19 \( 1 + (5.20 + 3.95i)T + (5.08 + 18.3i)T^{2} \)
23 \( 1 + (-3.46 - 6.54i)T + (-12.9 + 19.0i)T^{2} \)
29 \( 1 + (4.18 - 2.84i)T + (10.7 - 26.9i)T^{2} \)
31 \( 1 + (-3.65 - 4.80i)T + (-8.29 + 29.8i)T^{2} \)
37 \( 1 + (-1.74 - 7.94i)T + (-33.5 + 15.5i)T^{2} \)
41 \( 1 + (-7.32 - 3.88i)T + (23.0 + 33.9i)T^{2} \)
43 \( 1 + (3.07 + 3.24i)T + (-2.32 + 42.9i)T^{2} \)
47 \( 1 + (-5.64 + 2.61i)T + (30.4 - 35.8i)T^{2} \)
53 \( 1 + (-0.659 + 1.09i)T + (-24.8 - 46.8i)T^{2} \)
61 \( 1 + (-11.3 - 7.66i)T + (22.5 + 56.6i)T^{2} \)
67 \( 1 + (-2.90 + 13.1i)T + (-60.8 - 28.1i)T^{2} \)
71 \( 1 + (0.475 + 1.02i)T + (-45.9 + 54.1i)T^{2} \)
73 \( 1 + (1.72 - 5.13i)T + (-58.1 - 44.1i)T^{2} \)
79 \( 1 + (-0.278 + 5.12i)T + (-78.5 - 8.54i)T^{2} \)
83 \( 1 + (0.815 - 4.97i)T + (-78.6 - 26.5i)T^{2} \)
89 \( 1 + (2.39 + 3.52i)T + (-32.9 + 82.6i)T^{2} \)
97 \( 1 + (0.550 + 1.63i)T + (-77.2 + 58.7i)T^{2} \)
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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

−10.91500314250713461494115559690, −10.26973316092991916589504478717, −9.493784926605909439374520729927, −8.644420279336130257278589419999, −7.80284851110893965868864315773, −6.93323089854870419809959473491, −5.43899189149060176427635269577, −4.49029979923392207042572529173, −3.25844783931656701768200111902, −1.13566848555745272639184806836, 2.21225168072534585189359347239, 2.58335595383235614367055246407, 4.02099955508408293378891691307, 6.00572431501954471406283985588, 6.73691736207645411507181253203, 7.976580672941762085780342491838, 8.936499073316599603605765539753, 9.486466631413823632212155003546, 10.45652202434215509790213374917, 11.55854878512127632878607897654

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