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Properties

Label 2-336-84.11-c1-0-8
Degree $2$
Conductor $336$
Sign $0.963 - 0.266i$
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 + (0.5 − 2.59i)7-s + (1.5 + 2.59i)9-s + 7·13-s + (−4.5 + 2.59i)19-s + (3 − 3.46i)21-s + (−2.5 + 4.33i)25-s + 5.19i·27-s + (−7.5 − 4.33i)31-s + (0.5 + 0.866i)37-s + (10.5 + 6.06i)39-s − 12.1i·43-s + (−6.5 − 2.59i)49-s − 9·57-s + (−7 − 12.1i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (0.188 − 0.981i)7-s + (0.5 + 0.866i)9-s + 1.94·13-s + (−1.03 + 0.596i)19-s + (0.654 − 0.755i)21-s + (−0.5 + 0.866i)25-s + 0.999i·27-s + (−1.34 − 0.777i)31-s + (0.0821 + 0.142i)37-s + (1.68 + 0.970i)39-s − 1.84i·43-s + (−0.928 − 0.371i)49-s − 1.19·57-s + (−0.896 − 1.55i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.963 - 0.266i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.963 - 0.266i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80653 + 0.245465i\)
\(L(\frac12)\) \(\approx\) \(1.80653 + 0.245465i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 7T + 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.5 - 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (7.5 + 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 12.1iT - 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.5 - 6.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.5 - 6.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−11.16840852359451765119724649848, −10.73616364278287304953217711698, −9.733221932906258709951472011255, −8.710651413955247205602958665799, −8.015036189422552053944168052198, −6.95603385077621266170733743200, −5.63843513282617043377495200063, −4.10137997542137648858403093241, −3.59193014560405072238882888126, −1.72612189875925720973248939558, 1.67919957356378066769372483412, 2.98527943096114065119028565448, 4.21850698959596115677830892134, 5.86227972333237008020225043658, 6.64013876247236327421836003458, 8.005336318807721716619503830718, 8.666553837225620866397138182937, 9.292393667647362677410409950250, 10.67056682563100102171335089346, 11.56506801098004639560069405545

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