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Properties

Label 2-336-21.5-c1-0-8
Degree $2$
Conductor $336$
Sign $0.997 - 0.0633i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
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)3-s + (1.5 − 2.59i)5-s + (−2 + 1.73i)7-s + (1.5 + 2.59i)9-s + (4.5 − 2.59i)11-s + (4.5 − 2.59i)15-s + (1.5 + 2.59i)17-s + (−1.5 − 0.866i)19-s + (−4.5 + 0.866i)21-s + (−4.5 − 2.59i)23-s + (−2 − 3.46i)25-s + 5.19i·27-s + (1.5 − 0.866i)31-s + 9·33-s + (1.5 + 7.79i)35-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (0.670 − 1.16i)5-s + (−0.755 + 0.654i)7-s + (0.5 + 0.866i)9-s + (1.35 − 0.783i)11-s + (1.16 − 0.670i)15-s + (0.363 + 0.630i)17-s + (−0.344 − 0.198i)19-s + (−0.981 + 0.188i)21-s + (−0.938 − 0.541i)23-s + (−0.400 − 0.692i)25-s + 0.999i·27-s + (0.269 − 0.155i)31-s + 1.56·33-s + (0.253 + 1.31i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.997 - 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87143 + 0.0593210i\)
\(L(\frac12)\) \(\approx\) \(1.87143 + 0.0593210i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
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 \( 1 + (4.5 + 2.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 2.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.5 + 6.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−11.72806194118894479135695843276, −10.25553345651989378578977440504, −9.550247796574087908058448216093, −8.758466538063668562892171482394, −8.349791312818376349657542258306, −6.56856213515913679420664180465, −5.63398243039540336789581716427, −4.39258653545995870221884884757, −3.26166746578579665371705743328, −1.70130994460142571519624845944, 1.78856850140824550137597406758, 3.10728200225137527159652048523, 4.04931222030194004568255810021, 6.11451360904471004906158972989, 6.87709563233043355345867300694, 7.42183372786651225555018496260, 8.884719534279436096547994664889, 9.827847125976443015127326790840, 10.22605812207835988397233783971, 11.66823904581627870331971172577

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