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Properties

Label 2-336-48.35-c1-0-25
Degree $2$
Conductor $336$
Sign $0.987 + 0.156i$
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  + (1.41 + 0.0184i)2-s + (−1.52 − 0.827i)3-s + (1.99 + 0.0520i)4-s + (1.23 + 1.23i)5-s + (−2.13 − 1.19i)6-s − 7-s + (2.82 + 0.110i)8-s + (1.62 + 2.51i)9-s + (1.72 + 1.77i)10-s + (3.57 − 3.57i)11-s + (−2.99 − 1.73i)12-s + (1.58 + 1.58i)13-s + (−1.41 − 0.0184i)14-s + (−0.859 − 2.91i)15-s + (3.99 + 0.208i)16-s − 2.20i·17-s + ⋯
L(s)  = 1  + (0.999 + 0.0130i)2-s + (−0.878 − 0.477i)3-s + (0.999 + 0.0260i)4-s + (0.554 + 0.554i)5-s + (−0.872 − 0.489i)6-s − 0.377·7-s + (0.999 + 0.0390i)8-s + (0.543 + 0.839i)9-s + (0.546 + 0.561i)10-s + (1.07 − 1.07i)11-s + (−0.865 − 0.500i)12-s + (0.440 + 0.440i)13-s + (−0.377 − 0.00491i)14-s + (−0.221 − 0.751i)15-s + (0.998 + 0.0520i)16-s − 0.535i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.987 + 0.156i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.987 + 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07525 - 0.163872i\)
\(L(\frac12)\) \(\approx\) \(2.07525 - 0.163872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0184i)T \)
3 \( 1 + (1.52 + 0.827i)T \)
7 \( 1 + T \)
good5 \( 1 + (-1.23 - 1.23i)T + 5iT^{2} \)
11 \( 1 + (-3.57 + 3.57i)T - 11iT^{2} \)
13 \( 1 + (-1.58 - 1.58i)T + 13iT^{2} \)
17 \( 1 + 2.20iT - 17T^{2} \)
19 \( 1 + (3.76 - 3.76i)T - 19iT^{2} \)
23 \( 1 + 3.97iT - 23T^{2} \)
29 \( 1 + (4.75 - 4.75i)T - 29iT^{2} \)
31 \( 1 - 1.24iT - 31T^{2} \)
37 \( 1 + (6.39 - 6.39i)T - 37iT^{2} \)
41 \( 1 - 3.93T + 41T^{2} \)
43 \( 1 + (-1.19 - 1.19i)T + 43iT^{2} \)
47 \( 1 + 8.26T + 47T^{2} \)
53 \( 1 + (2.18 + 2.18i)T + 53iT^{2} \)
59 \( 1 + (5.45 - 5.45i)T - 59iT^{2} \)
61 \( 1 + (5.10 + 5.10i)T + 61iT^{2} \)
67 \( 1 + (7.23 - 7.23i)T - 67iT^{2} \)
71 \( 1 - 6.13iT - 71T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 + 6.53iT - 79T^{2} \)
83 \( 1 + (12.3 + 12.3i)T + 83iT^{2} \)
89 \( 1 + 9.15T + 89T^{2} \)
97 \( 1 - 7.67T + 97T^{2} \)
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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

−11.62695891632448141082186035097, −10.89592965762723818522528498689, −10.13386358881538650806778628979, −8.610902976122294797037315380658, −7.16233474065334875158138817242, −6.28272861684734651757681158529, −6.01064543531830851482742238313, −4.57198479870019243676051964028, −3.27528136142552950375518179034, −1.68835899872111232322271524024, 1.69641706137171938469234688921, 3.70135024189540865287851007706, 4.57499765731507975955051412474, 5.62707401039098295253390133234, 6.35686173674436775796356671977, 7.32539719291327674203419542129, 9.100917403586447685164558292589, 9.872411674796305495881757985133, 10.89406498836210529545591424422, 11.64934722220900962944775207209

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