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Properties

Label 2-351-117.25-c1-0-8
Degree $2$
Conductor $351$
Sign $0.316 - 0.948i$
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  + (2.14 + 1.23i)2-s + (2.06 + 3.58i)4-s + (0.771 − 0.445i)5-s + (0.850 + 0.491i)7-s + 5.29i·8-s + 2.20·10-s + (−2.49 − 1.44i)11-s + (0.714 − 3.53i)13-s + (1.21 + 2.10i)14-s + (−2.42 + 4.19i)16-s − 5.34·17-s + 7.19i·19-s + (3.19 + 1.84i)20-s + (−3.57 − 6.19i)22-s + (−2.31 − 4.01i)23-s + ⋯
L(s)  = 1  + (1.51 + 0.875i)2-s + (1.03 + 1.79i)4-s + (0.344 − 0.199i)5-s + (0.321 + 0.185i)7-s + 1.87i·8-s + 0.697·10-s + (−0.753 − 0.435i)11-s + (0.198 − 0.980i)13-s + (0.325 + 0.563i)14-s + (−0.605 + 1.04i)16-s − 1.29·17-s + 1.65i·19-s + (0.713 + 0.411i)20-s + (−0.762 − 1.31i)22-s + (−0.483 − 0.836i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.316 - 0.948i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45032 + 1.76462i\)
\(L(\frac12)\) \(\approx\) \(2.45032 + 1.76462i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.714 + 3.53i)T \)
good2 \( 1 + (-2.14 - 1.23i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.771 + 0.445i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.850 - 0.491i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.49 + 1.44i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.34T + 17T^{2} \)
19 \( 1 - 7.19iT - 19T^{2} \)
23 \( 1 + (2.31 + 4.01i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.971 + 1.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.73 + 5.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.82iT - 37T^{2} \)
41 \( 1 + (2.39 - 1.38i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.45 - 4.25i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.82 - 2.20i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.30T + 53T^{2} \)
59 \( 1 + (-2.74 + 1.58i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.76 + 4.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.15 - 1.81i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.69iT - 71T^{2} \)
73 \( 1 - 9.33iT - 73T^{2} \)
79 \( 1 + (2.37 - 4.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.78 - 2.76i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 17.5iT - 89T^{2} \)
97 \( 1 + (-0.213 - 0.123i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.02727782027278306868498765513, −11.02686046462528949409226557838, −9.924143931402390969248603397679, −8.301113186829522433668924299736, −7.85733497186873562392508439998, −6.42616751013848853134384883057, −5.77775402176255171906474490584, −4.89729364993113462135140369402, −3.79727188875150930477653494421, −2.47875899229112291077604061235, 1.89949062720194689835239479290, 2.88532222155371967342881328152, 4.38816254785374672570767011895, 4.90315646433088657341764499739, 6.22937626416128735302384304024, 7.03578643506528883121157786415, 8.650280490705993747555134246675, 9.903314356642529195390644268699, 10.72319047613120331291788041129, 11.48301635912661978117995041658

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