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Properties

Label 2-351-117.49-c1-0-5
Degree $2$
Conductor $351$
Sign $0.568 - 0.822i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + (1.5 + 0.866i)2-s + (0.5 + 0.866i)4-s + (1.5 + 0.866i)5-s + 1.73i·7-s − 1.73i·8-s + (1.5 + 2.59i)10-s + (3 + 1.73i)11-s + (−2.5 + 2.59i)13-s + (−1.49 + 2.59i)14-s + (2.49 − 4.33i)16-s + (1.5 − 2.59i)17-s + (−1.5 − 0.866i)19-s + 1.73i·20-s + (3 + 5.19i)22-s + 3·23-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + (0.250 + 0.433i)4-s + (0.670 + 0.387i)5-s + 0.654i·7-s − 0.612i·8-s + (0.474 + 0.821i)10-s + (0.904 + 0.522i)11-s + (−0.693 + 0.720i)13-s + (−0.400 + 0.694i)14-s + (0.624 − 1.08i)16-s + (0.363 − 0.630i)17-s + (−0.344 − 0.198i)19-s + 0.387i·20-s + (0.639 + 1.10i)22-s + 0.625·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.568 - 0.822i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.568 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15274 + 1.12943i\)
\(L(\frac12)\) \(\approx\) \(2.15274 + 1.12943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (2.5 - 2.59i)T \)
good2 \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.5 + 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.5 - 2.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 12.1iT - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-4.5 + 2.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (3 - 1.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 5T + 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + (-7.5 - 4.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 2.59i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (13.5 - 7.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.5iT - 97T^{2} \)
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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

−12.06272799270589081783280802712, −10.76702274561488779772621948578, −9.557007754468924166758341711479, −9.108177828379414905144917231309, −7.30128132479318039200971719674, −6.70140186094699581459740289206, −5.69737991862035844694998217789, −4.87532785684935024652013583685, −3.66478645975281470680206126249, −2.13139590154041162468007862473, 1.62384645393228916400991854501, 3.18463874775291746987312162959, 4.15659487332322069359225162815, 5.27873209964208791147404669402, 6.09445080528682350553062460267, 7.50099712722006487780172718095, 8.634735120985183765114873176890, 9.674193346269544006740892761819, 10.68339556572670650017439552068, 11.44426157986036709761345784037

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