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Properties

Label 2-336-48.11-c1-0-31
Degree $2$
Conductor $336$
Sign $0.253 + 0.967i$
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  + (−0.724 − 1.21i)2-s + (1.72 − 0.119i)3-s + (−0.950 + 1.75i)4-s + (0.793 − 0.793i)5-s + (−1.39 − 2.01i)6-s − 7-s + (2.82 − 0.119i)8-s + (2.97 − 0.414i)9-s + (−1.53 − 0.389i)10-s + (−0.687 − 0.687i)11-s + (−1.43 + 3.15i)12-s + (2.91 − 2.91i)13-s + (0.724 + 1.21i)14-s + (1.27 − 1.46i)15-s + (−2.19 − 3.34i)16-s − 1.81i·17-s + ⋯
L(s)  = 1  + (−0.512 − 0.858i)2-s + (0.997 − 0.0691i)3-s + (−0.475 + 0.879i)4-s + (0.354 − 0.354i)5-s + (−0.570 − 0.821i)6-s − 0.377·7-s + (0.999 − 0.0423i)8-s + (0.990 − 0.138i)9-s + (−0.486 − 0.123i)10-s + (−0.207 − 0.207i)11-s + (−0.413 + 0.910i)12-s + (0.807 − 0.807i)13-s + (0.193 + 0.324i)14-s + (0.329 − 0.378i)15-s + (−0.548 − 0.836i)16-s − 0.441i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.253 + 0.967i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.253 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14255 - 0.881425i\)
\(L(\frac12)\) \(\approx\) \(1.14255 - 0.881425i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.724 + 1.21i)T \)
3 \( 1 + (-1.72 + 0.119i)T \)
7 \( 1 + T \)
good5 \( 1 + (-0.793 + 0.793i)T - 5iT^{2} \)
11 \( 1 + (0.687 + 0.687i)T + 11iT^{2} \)
13 \( 1 + (-2.91 + 2.91i)T - 13iT^{2} \)
17 \( 1 + 1.81iT - 17T^{2} \)
19 \( 1 + (-3.24 - 3.24i)T + 19iT^{2} \)
23 \( 1 + 5.92iT - 23T^{2} \)
29 \( 1 + (-1.54 - 1.54i)T + 29iT^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 + (5.10 + 5.10i)T + 37iT^{2} \)
41 \( 1 - 1.19T + 41T^{2} \)
43 \( 1 + (7.00 - 7.00i)T - 43iT^{2} \)
47 \( 1 + 13.1T + 47T^{2} \)
53 \( 1 + (2.08 - 2.08i)T - 53iT^{2} \)
59 \( 1 + (0.919 + 0.919i)T + 59iT^{2} \)
61 \( 1 + (-4.21 + 4.21i)T - 61iT^{2} \)
67 \( 1 + (-7.87 - 7.87i)T + 67iT^{2} \)
71 \( 1 - 7.99iT - 71T^{2} \)
73 \( 1 - 4.19iT - 73T^{2} \)
79 \( 1 - 8.07iT - 79T^{2} \)
83 \( 1 + (11.0 - 11.0i)T - 83iT^{2} \)
89 \( 1 - 7.73T + 89T^{2} \)
97 \( 1 - 6.61T + 97T^{2} \)
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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

−11.20833289739615819325406691702, −10.19996005216263461311001861040, −9.582066299589166018021269825849, −8.604793614535472987862665331230, −8.049136623951668556098456800862, −6.83114653682465143901667297727, −5.15678179713231119789956889647, −3.68923252418505261907933206960, −2.84483501031508035497528059176, −1.33048976716764653091008112050, 1.80350204748554737819279403324, 3.48995827231403267347694916161, 4.84015590144542717882820065253, 6.25574954705163334146630706770, 7.03076373572577718230407494221, 8.029687907897856488348990072026, 8.904695497159084972775856019977, 9.691579179182917306972888253494, 10.33793296026996029338804383767, 11.58320743576505170352781874196

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