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Properties

Label 2-336-16.5-c1-0-12
Degree $2$
Conductor $336$
Sign $0.883 + 0.469i$
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  + (−1.19 + 0.757i)2-s + (0.707 − 0.707i)3-s + (0.852 − 1.80i)4-s + (−0.894 − 0.894i)5-s + (−0.309 + 1.38i)6-s + i·7-s + (0.351 + 2.80i)8-s − 1.00i·9-s + (1.74 + 0.390i)10-s + (2.02 + 2.02i)11-s + (−0.676 − 1.88i)12-s + (3.23 − 3.23i)13-s + (−0.757 − 1.19i)14-s − 1.26·15-s + (−2.54 − 3.08i)16-s + 0.119·17-s + ⋯
L(s)  = 1  + (−0.844 + 0.535i)2-s + (0.408 − 0.408i)3-s + (0.426 − 0.904i)4-s + (−0.399 − 0.399i)5-s + (−0.126 + 0.563i)6-s + 0.377i·7-s + (0.124 + 0.992i)8-s − 0.333i·9-s + (0.551 + 0.123i)10-s + (0.610 + 0.610i)11-s + (−0.195 − 0.543i)12-s + (0.896 − 0.896i)13-s + (−0.202 − 0.319i)14-s − 0.326·15-s + (−0.636 − 0.771i)16-s + 0.0290·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.883 + 0.469i$
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: \((2,\ 336,\ (\ :1/2),\ 0.883 + 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.974589 - 0.242874i\)
\(L(\frac12)\) \(\approx\) \(0.974589 - 0.242874i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 - 0.757i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 - iT \)
good5 \( 1 + (0.894 + 0.894i)T + 5iT^{2} \)
11 \( 1 + (-2.02 - 2.02i)T + 11iT^{2} \)
13 \( 1 + (-3.23 + 3.23i)T - 13iT^{2} \)
17 \( 1 - 0.119T + 17T^{2} \)
19 \( 1 + (-4.85 + 4.85i)T - 19iT^{2} \)
23 \( 1 + 9.33iT - 23T^{2} \)
29 \( 1 + (5.18 - 5.18i)T - 29iT^{2} \)
31 \( 1 - 0.957T + 31T^{2} \)
37 \( 1 + (-0.136 - 0.136i)T + 37iT^{2} \)
41 \( 1 - 3.36iT - 41T^{2} \)
43 \( 1 + (-8.82 - 8.82i)T + 43iT^{2} \)
47 \( 1 - 3.50T + 47T^{2} \)
53 \( 1 + (5.46 + 5.46i)T + 53iT^{2} \)
59 \( 1 + (-5.48 - 5.48i)T + 59iT^{2} \)
61 \( 1 + (8.06 - 8.06i)T - 61iT^{2} \)
67 \( 1 + (8.09 - 8.09i)T - 67iT^{2} \)
71 \( 1 - 3.42iT - 71T^{2} \)
73 \( 1 - 2.48iT - 73T^{2} \)
79 \( 1 + 6.89T + 79T^{2} \)
83 \( 1 + (-11.1 + 11.1i)T - 83iT^{2} \)
89 \( 1 - 3.31iT - 89T^{2} \)
97 \( 1 + 5.52T + 97T^{2} \)
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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

−11.39079163291046753985192294248, −10.41020962926548822996967514281, −9.256765746506117120798628285524, −8.670456173309688985623168756148, −7.81173411451715896619985780959, −6.88657861334012280012729090154, −5.88442113639404765007263904785, −4.57378269636066060527492635230, −2.74291656102580372261498846493, −1.03399427121434499818018534039, 1.57726347634880584619816736734, 3.46710513114992282046816341573, 3.85233278525033302193291860872, 5.90339594481822768167356786071, 7.27740751803247522820907832907, 7.901979193884082960087102890159, 9.130415789913924847719074168431, 9.571061091741832418832824979648, 10.80141988466469196326554829226, 11.36208377662703111362743719410

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