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Properties

Label 2-354-177.176-c3-0-42
Degree $2$
Conductor $354$
Sign $0.896 - 0.444i$
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 + (5.02 − 1.30i)3-s + 4·4-s + 12.2i·5-s + (10.0 − 2.60i)6-s + 29.7·7-s + 8·8-s + (23.5 − 13.1i)9-s + 24.5i·10-s − 28.5·11-s + (20.1 − 5.21i)12-s + 60.2i·13-s + 59.5·14-s + (16.0 + 61.7i)15-s + 16·16-s − 55.9i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.967 − 0.250i)3-s + 0.5·4-s + 1.09i·5-s + (0.684 − 0.177i)6-s + 1.60·7-s + 0.353·8-s + (0.874 − 0.485i)9-s + 0.776i·10-s − 0.781·11-s + (0.483 − 0.125i)12-s + 1.28i·13-s + 1.13·14-s + (0.275 + 1.06i)15-s + 0.250·16-s − 0.798i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.444i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.896 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.896 - 0.444i$
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: \((2,\ 354,\ (\ :3/2),\ 0.896 - 0.444i)\)

Particular Values

\(L(2)\) \(\approx\) \(4.647952631\)
\(L(\frac12)\) \(\approx\) \(4.647952631\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.02 + 1.30i)T \)
59 \( 1 + (443. - 92.8i)T \)
good5 \( 1 - 12.2iT - 125T^{2} \)
7 \( 1 - 29.7T + 343T^{2} \)
11 \( 1 + 28.5T + 1.33e3T^{2} \)
13 \( 1 - 60.2iT - 2.19e3T^{2} \)
17 \( 1 + 55.9iT - 4.91e3T^{2} \)
19 \( 1 + 75.7T + 6.85e3T^{2} \)
23 \( 1 + 54.0T + 1.21e4T^{2} \)
29 \( 1 - 189. iT - 2.43e4T^{2} \)
31 \( 1 + 279. iT - 2.97e4T^{2} \)
37 \( 1 - 82.3iT - 5.06e4T^{2} \)
41 \( 1 + 211. iT - 6.89e4T^{2} \)
43 \( 1 - 262. iT - 7.95e4T^{2} \)
47 \( 1 - 410.T + 1.03e5T^{2} \)
53 \( 1 + 436. iT - 1.48e5T^{2} \)
61 \( 1 + 745. iT - 2.26e5T^{2} \)
67 \( 1 - 317. iT - 3.00e5T^{2} \)
71 \( 1 - 61.7iT - 3.57e5T^{2} \)
73 \( 1 + 912. iT - 3.89e5T^{2} \)
79 \( 1 + 49.0T + 4.93e5T^{2} \)
83 \( 1 - 156.T + 5.71e5T^{2} \)
89 \( 1 + 901.T + 7.04e5T^{2} \)
97 \( 1 + 20.3iT - 9.12e5T^{2} \)
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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

−11.17552672065881384723146742242, −10.44725607281520081791359252610, −9.148352999640322194828648972609, −8.030017259155016892633453585041, −7.35622730093179459114611947515, −6.45797109011716799648007628435, −4.93209774876470302944544576958, −3.99014549909187879734711880946, −2.61642733536201334778736979215, −1.85047592263594305581034817752, 1.36607372359089783677237769417, 2.54023179898482874752872892129, 4.09139289302306945969254936733, 4.83524390844006829622223459930, 5.66922106337954794168662353997, 7.54427445582255795575257270335, 8.227428239002211522680938597552, 8.721847246540440543331719404721, 10.29040035226324088782746227216, 10.83276127161544682667652174346

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