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Properties

Label 2-354-177.176-c3-0-19
Degree $2$
Conductor $354$
Sign $0.603 - 0.797i$
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 + (−3.04 + 4.21i)3-s + 4·4-s − 12.5i·5-s + (−6.08 + 8.42i)6-s − 33.4·7-s + 8·8-s + (−8.51 − 25.6i)9-s − 25.0i·10-s + 55.6·11-s + (−12.1 + 16.8i)12-s + 53.4i·13-s − 66.8·14-s + (52.8 + 38.1i)15-s + 16·16-s + 114. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.585 + 0.810i)3-s + 0.5·4-s − 1.12i·5-s + (−0.413 + 0.573i)6-s − 1.80·7-s + 0.353·8-s + (−0.315 − 0.949i)9-s − 0.793i·10-s + 1.52·11-s + (−0.292 + 0.405i)12-s + 1.14i·13-s − 1.27·14-s + (0.909 + 0.656i)15-s + 0.250·16-s + 1.63i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.603 - 0.797i$
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: \((2,\ 354,\ (\ :3/2),\ 0.603 - 0.797i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.006678413\)
\(L(\frac12)\) \(\approx\) \(2.006678413\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.04 - 4.21i)T \)
59 \( 1 + (-453. - 10.4i)T \)
good5 \( 1 + 12.5iT - 125T^{2} \)
7 \( 1 + 33.4T + 343T^{2} \)
11 \( 1 - 55.6T + 1.33e3T^{2} \)
13 \( 1 - 53.4iT - 2.19e3T^{2} \)
17 \( 1 - 114. iT - 4.91e3T^{2} \)
19 \( 1 - 78.3T + 6.85e3T^{2} \)
23 \( 1 - 163.T + 1.21e4T^{2} \)
29 \( 1 + 76.0iT - 2.43e4T^{2} \)
31 \( 1 - 160. iT - 2.97e4T^{2} \)
37 \( 1 - 83.0iT - 5.06e4T^{2} \)
41 \( 1 + 70.9iT - 6.89e4T^{2} \)
43 \( 1 + 257. iT - 7.95e4T^{2} \)
47 \( 1 + 497.T + 1.03e5T^{2} \)
53 \( 1 - 500. iT - 1.48e5T^{2} \)
61 \( 1 + 32.9iT - 2.26e5T^{2} \)
67 \( 1 + 1.05e3iT - 3.00e5T^{2} \)
71 \( 1 - 470. iT - 3.57e5T^{2} \)
73 \( 1 - 912. iT - 3.89e5T^{2} \)
79 \( 1 - 674.T + 4.93e5T^{2} \)
83 \( 1 - 483.T + 5.71e5T^{2} \)
89 \( 1 - 70.2T + 7.04e5T^{2} \)
97 \( 1 - 549. iT - 9.12e5T^{2} \)
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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

−11.40324452101975586483624738831, −10.21046846090894202180925566435, −9.281586109067991264543309844595, −8.886734954893947154470083639864, −6.74953901801816676269767511316, −6.31860433847833618445455711118, −5.16733015607623970787041112639, −4.07256919378968888373353177033, −3.43154282600335321686803325321, −1.13853067845949574988401532837, 0.73069446436582033498155369542, 2.82275152888177339977975676325, 3.35467088435517006698513716809, 5.21221531507597337966949177679, 6.32323452236465709281409886275, 6.80331863497483468394783056300, 7.44497972316297648143188632969, 9.280485469032885405442401976880, 10.13594974371133296698465054144, 11.26241837963138574490320472028

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