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Properties

Label 2-336-48.35-c1-0-47
Degree $2$
Conductor $336$
Sign $-0.993 - 0.117i$
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 + (−0.119 − 1.72i)3-s + (−0.950 − 1.75i)4-s + (−0.793 − 0.793i)5-s + (−2.18 − 1.10i)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 + (−2.92 + 1.85i)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 + (−0.0691 − 0.997i)3-s + (−0.475 − 0.879i)4-s + (−0.354 − 0.354i)5-s + (−0.892 − 0.451i)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 + (−0.844 + 0.535i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.993 - 0.117i$
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: \((2,\ 336,\ (\ :1/2),\ -0.993 - 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0739191 + 1.24966i\)
\(L(\frac12)\) \(\approx\) \(0.0739191 + 1.24966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.119 + 1.72i)T \)
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

−11.45696498524193382831495972705, −10.43077264324450768257927477806, −9.110105295481341477647875427847, −8.499791323254509599242596029159, −6.99877646653815736643902546413, −6.16578612232567527700554089822, −4.96677156316074381993108325643, −3.66583644424518246540158551161, −2.35022458746941016024459843528, −0.78198810646755960366245490275, 3.32547637125642759844221483816, 3.80889398752449458479241277579, 5.27608630780174194021117672326, 5.96044540885714020629741041163, 7.22340570901653917946695914186, 8.204529663657355047874911711566, 9.161998096594124528627511781147, 10.09298377747580734983046736922, 11.16329429101377276638398938283, 12.02139774131145076069051311540

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