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Properties

Label 2-354-177.176-c3-0-56
Degree $2$
Conductor $354$
Sign $-0.946 - 0.323i$
Analytic cond. $20.8866$
Root an. cond. $4.57019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + (3.65 − 3.69i)3-s + 4·4-s − 2.23i·5-s + (−7.31 + 7.38i)6-s + 2.58·7-s − 8·8-s + (−0.280 − 26.9i)9-s + 4.46i·10-s − 43.0·11-s + (14.6 − 14.7i)12-s + 25.4i·13-s − 5.17·14-s + (−8.23 − 8.15i)15-s + 16·16-s − 0.642i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.703 − 0.710i)3-s + 0.5·4-s − 0.199i·5-s + (−0.497 + 0.502i)6-s + 0.139·7-s − 0.353·8-s + (−0.0103 − 0.999i)9-s + 0.141i·10-s − 1.18·11-s + (0.351 − 0.355i)12-s + 0.543i·13-s − 0.0988·14-s + (−0.141 − 0.140i)15-s + 0.250·16-s − 0.00916i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.946 - 0.323i$
Analytic conductor: \(20.8866\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :3/2),\ -0.946 - 0.323i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3561079451\)
\(L(\frac12)\) \(\approx\) \(0.3561079451\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 + (-3.65 + 3.69i)T \)
59 \( 1 + (197. + 407. i)T \)
good5 \( 1 + 2.23iT - 125T^{2} \)
7 \( 1 - 2.58T + 343T^{2} \)
11 \( 1 + 43.0T + 1.33e3T^{2} \)
13 \( 1 - 25.4iT - 2.19e3T^{2} \)
17 \( 1 + 0.642iT - 4.91e3T^{2} \)
19 \( 1 + 139.T + 6.85e3T^{2} \)
23 \( 1 + 132.T + 1.21e4T^{2} \)
29 \( 1 + 166. iT - 2.43e4T^{2} \)
31 \( 1 - 140. iT - 2.97e4T^{2} \)
37 \( 1 + 338. iT - 5.06e4T^{2} \)
41 \( 1 - 296. iT - 6.89e4T^{2} \)
43 \( 1 - 207. iT - 7.95e4T^{2} \)
47 \( 1 - 54.5T + 1.03e5T^{2} \)
53 \( 1 - 27.5iT - 1.48e5T^{2} \)
61 \( 1 - 474. iT - 2.26e5T^{2} \)
67 \( 1 + 525. iT - 3.00e5T^{2} \)
71 \( 1 - 977. iT - 3.57e5T^{2} \)
73 \( 1 + 707. iT - 3.89e5T^{2} \)
79 \( 1 + 629.T + 4.93e5T^{2} \)
83 \( 1 + 776.T + 5.71e5T^{2} \)
89 \( 1 + 1.43e3T + 7.04e5T^{2} \)
97 \( 1 - 169. iT - 9.12e5T^{2} \)
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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

−10.41109558072012330597820473200, −9.450075859610362739701072329665, −8.449886665205043347265629071623, −7.979473881876616825338566264961, −6.91266151630320311079300622824, −5.96837653650615583838238103764, −4.33244632126467878148831626904, −2.74842381553887207252282339638, −1.77024578326442371595412513831, −0.13018389474859530231036781892, 2.08460289956029222804294696182, 3.11270017990151191632789592800, 4.49927128933195292933942358371, 5.70460494884622989784861199142, 7.09392525110712839372869642439, 8.192047957461301265779014246349, 8.572702972869983723530507515636, 9.832894329510891902918526995710, 10.48863413019382714062406119164, 11.02229898482421563875151449639

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