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Properties

Label 2-354-177.176-c3-0-20
Degree $2$
Conductor $354$
Sign $-0.927 - 0.373i$
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 + (3.01 + 4.23i)3-s + 4·4-s + 19.8i·5-s + (−6.02 − 8.46i)6-s + 26.1·7-s − 8·8-s + (−8.82 + 25.5i)9-s − 39.6i·10-s − 40.5·11-s + (12.0 + 16.9i)12-s + 74.9i·13-s − 52.3·14-s + (−83.8 + 59.6i)15-s + 16·16-s − 1.85i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.580 + 0.814i)3-s + 0.5·4-s + 1.77i·5-s + (−0.410 − 0.575i)6-s + 1.41·7-s − 0.353·8-s + (−0.327 + 0.945i)9-s − 1.25i·10-s − 1.11·11-s + (0.290 + 0.407i)12-s + 1.59i·13-s − 0.998·14-s + (−1.44 + 1.02i)15-s + 0.250·16-s − 0.0265i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.927 - 0.373i$
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: \((2,\ 354,\ (\ :3/2),\ -0.927 - 0.373i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.731888735\)
\(L(\frac12)\) \(\approx\) \(1.731888735\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.01 - 4.23i)T \)
59 \( 1 + (381. - 244. i)T \)
good5 \( 1 - 19.8iT - 125T^{2} \)
7 \( 1 - 26.1T + 343T^{2} \)
11 \( 1 + 40.5T + 1.33e3T^{2} \)
13 \( 1 - 74.9iT - 2.19e3T^{2} \)
17 \( 1 + 1.85iT - 4.91e3T^{2} \)
19 \( 1 - 80.9T + 6.85e3T^{2} \)
23 \( 1 - 176.T + 1.21e4T^{2} \)
29 \( 1 + 227. iT - 2.43e4T^{2} \)
31 \( 1 - 52.4iT - 2.97e4T^{2} \)
37 \( 1 + 275. iT - 5.06e4T^{2} \)
41 \( 1 + 132. iT - 6.89e4T^{2} \)
43 \( 1 - 85.9iT - 7.95e4T^{2} \)
47 \( 1 - 503.T + 1.03e5T^{2} \)
53 \( 1 + 277. iT - 1.48e5T^{2} \)
61 \( 1 - 770. iT - 2.26e5T^{2} \)
67 \( 1 + 84.3iT - 3.00e5T^{2} \)
71 \( 1 + 188. iT - 3.57e5T^{2} \)
73 \( 1 + 169. iT - 3.89e5T^{2} \)
79 \( 1 + 674.T + 4.93e5T^{2} \)
83 \( 1 + 47.0T + 5.71e5T^{2} \)
89 \( 1 + 466.T + 7.04e5T^{2} \)
97 \( 1 - 577. iT - 9.12e5T^{2} \)
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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

−11.08699882702028390641624991069, −10.61733017591953126131482184041, −9.674399429412496794150062796296, −8.730780955610419596095062244213, −7.61064662735860092776736135577, −7.17591587619298188637870345125, −5.60216017282583690206777727438, −4.30260781154432584856259605114, −2.90664950409021571964173750108, −2.06673098254842790679904100705, 0.75733935374510750052662402660, 1.44718348242414579260039927939, 2.95583760820473103092866738156, 4.96057060973993865038300028304, 5.51024703463155426245499887506, 7.40873893196561298148495140378, 8.026041091841724265987585851332, 8.522607235257423950625673176139, 9.344657661698287215869528672572, 10.61220347119122257723466758704

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