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Properties

Label 2-351-117.22-c1-0-5
Degree $2$
Conductor $351$
Sign $-0.680 + 0.732i$
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  − 0.697·2-s − 1.51·4-s + (−1.44 + 2.50i)5-s + (1.58 − 2.74i)7-s + 2.45·8-s + (1.00 − 1.74i)10-s − 2.31·11-s + (−3.15 + 1.74i)13-s + (−1.10 + 1.91i)14-s + 1.31·16-s + (−2.69 − 4.66i)17-s + (−2.58 − 4.48i)19-s + (2.18 − 3.78i)20-s + 1.61·22-s + (−3.27 − 5.66i)23-s + ⋯
L(s)  = 1  − 0.493·2-s − 0.756·4-s + (−0.646 + 1.11i)5-s + (0.599 − 1.03i)7-s + 0.866·8-s + (0.318 − 0.552i)10-s − 0.697·11-s + (−0.874 + 0.484i)13-s + (−0.295 + 0.512i)14-s + 0.329·16-s + (−0.653 − 1.13i)17-s + (−0.593 − 1.02i)19-s + (0.489 − 0.847i)20-s + 0.343·22-s + (−0.682 − 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.680 + 0.732i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.680 + 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0967788 - 0.221958i\)
\(L(\frac12)\) \(\approx\) \(0.0967788 - 0.221958i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.15 - 1.74i)T \)
good2 \( 1 + 0.697T + 2T^{2} \)
5 \( 1 + (1.44 - 2.50i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.58 + 2.74i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.31T + 11T^{2} \)
17 \( 1 + (2.69 + 4.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.58 + 4.48i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.27 + 5.66i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.02T + 29T^{2} \)
31 \( 1 + (-4.23 + 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.42 - 4.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.25 - 2.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.99 - 5.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.521 + 0.902i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.29T + 53T^{2} \)
59 \( 1 - 4.70T + 59T^{2} \)
61 \( 1 + (3.71 - 6.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.18 + 7.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.680 - 1.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + (0.0365 + 0.0633i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.08 - 1.88i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.0891 + 0.154i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0654 - 0.113i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.95087934131835609097871955469, −10.30585743825819458618825396782, −9.388690523354714626216737346975, −8.156351228594031788508563427644, −7.47484832138338747378558453491, −6.73307561798293068702659275341, −4.81178055494986423332812271307, −4.20001640931380776677180264624, −2.58005756946408701739276668734, −0.19715882142248416146662610022, 1.81991496191392049256682899905, 3.90895305116516165445001140589, 4.97013920026392269139537106640, 5.63550578391304035159275107042, 7.60876819004339582221092413701, 8.362289425981901881013923928906, 8.726872407557133584602135298399, 9.847408148610641129274938429751, 10.75137941343724097063891444918, 12.09021863073290868375117981453

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