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Properties

Label 2-336-16.5-c1-0-19
Degree $2$
Conductor $336$
Sign $0.0537 + 0.998i$
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  + (0.428 + 1.34i)2-s + (−0.707 + 0.707i)3-s + (−1.63 + 1.15i)4-s + (−1.77 − 1.77i)5-s + (−1.25 − 0.649i)6-s − i·7-s + (−2.25 − 1.70i)8-s − 1.00i·9-s + (1.62 − 3.14i)10-s + (−3.99 − 3.99i)11-s + (0.337 − 1.97i)12-s + (−2.40 + 2.40i)13-s + (1.34 − 0.428i)14-s + 2.50·15-s + (1.32 − 3.77i)16-s + 3.97·17-s + ⋯
L(s)  = 1  + (0.303 + 0.952i)2-s + (−0.408 + 0.408i)3-s + (−0.816 + 0.577i)4-s + (−0.792 − 0.792i)5-s + (−0.512 − 0.265i)6-s − 0.377i·7-s + (−0.797 − 0.602i)8-s − 0.333i·9-s + (0.515 − 0.995i)10-s + (−1.20 − 1.20i)11-s + (0.0973 − 0.569i)12-s + (−0.667 + 0.667i)13-s + (0.360 − 0.114i)14-s + 0.647·15-s + (0.332 − 0.943i)16-s + 0.963·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0537 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0537 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.0537 + 0.998i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.0537 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180442 - 0.170990i\)
\(L(\frac12)\) \(\approx\) \(0.180442 - 0.170990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.428 - 1.34i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + iT \)
good5 \( 1 + (1.77 + 1.77i)T + 5iT^{2} \)
11 \( 1 + (3.99 + 3.99i)T + 11iT^{2} \)
13 \( 1 + (2.40 - 2.40i)T - 13iT^{2} \)
17 \( 1 - 3.97T + 17T^{2} \)
19 \( 1 + (3.91 - 3.91i)T - 19iT^{2} \)
23 \( 1 + 1.97iT - 23T^{2} \)
29 \( 1 + (-2.25 + 2.25i)T - 29iT^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + (2.76 + 2.76i)T + 37iT^{2} \)
41 \( 1 - 5.44iT - 41T^{2} \)
43 \( 1 + (-7.09 - 7.09i)T + 43iT^{2} \)
47 \( 1 + 6.40T + 47T^{2} \)
53 \( 1 + (8.01 + 8.01i)T + 53iT^{2} \)
59 \( 1 + (4.43 + 4.43i)T + 59iT^{2} \)
61 \( 1 + (-4.80 + 4.80i)T - 61iT^{2} \)
67 \( 1 + (10.0 - 10.0i)T - 67iT^{2} \)
71 \( 1 + 3.11iT - 71T^{2} \)
73 \( 1 + 3.22iT - 73T^{2} \)
79 \( 1 - 9.44T + 79T^{2} \)
83 \( 1 + (-3.33 + 3.33i)T - 83iT^{2} \)
89 \( 1 - 7.05iT - 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−11.44621135127957022786366032629, −10.37322349742876024070568527731, −9.263854713188505139391496621736, −8.192399021409020531867685047818, −7.72105201449970218645130429370, −6.31601063462922630057745746573, −5.32242713402742100641963744970, −4.47258434166732952188252096783, −3.44830202738742558692506534984, −0.15972804639709321761849158489, 2.20465017212977791864757468447, 3.27402016482905585366038275926, 4.75856074583590433348833577755, 5.60174524851894316743525408630, 7.14492500353818477018235299482, 7.83276507603457072352992702705, 9.225195383429990330187252421853, 10.44610652873111361165359015479, 10.76509611291710105153819244124, 11.89945594801152763596832353346

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