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Properties

Label 2-336-16.5-c1-0-17
Degree $2$
Conductor $336$
Sign $0.973 + 0.228i$
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  + (1.39 + 0.219i)2-s + (0.707 − 0.707i)3-s + (1.90 + 0.613i)4-s + (−0.539 − 0.539i)5-s + (1.14 − 0.832i)6-s − i·7-s + (2.52 + 1.27i)8-s − 1.00i·9-s + (−0.635 − 0.871i)10-s + (0.697 + 0.697i)11-s + (1.77 − 0.912i)12-s + (−0.00866 + 0.00866i)13-s + (0.219 − 1.39i)14-s − 0.762·15-s + (3.24 + 2.33i)16-s − 2.70·17-s + ⋯
L(s)  = 1  + (0.987 + 0.155i)2-s + (0.408 − 0.408i)3-s + (0.951 + 0.306i)4-s + (−0.241 − 0.241i)5-s + (0.466 − 0.339i)6-s − 0.377i·7-s + (0.892 + 0.450i)8-s − 0.333i·9-s + (−0.200 − 0.275i)10-s + (0.210 + 0.210i)11-s + (0.513 − 0.263i)12-s + (−0.00240 + 0.00240i)13-s + (0.0586 − 0.373i)14-s − 0.196·15-s + (0.811 + 0.583i)16-s − 0.656·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.973 + 0.228i$
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: \((2,\ 336,\ (\ :1/2),\ 0.973 + 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56264 - 0.296797i\)
\(L(\frac12)\) \(\approx\) \(2.56264 - 0.296797i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.219i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + iT \)
good5 \( 1 + (0.539 + 0.539i)T + 5iT^{2} \)
11 \( 1 + (-0.697 - 0.697i)T + 11iT^{2} \)
13 \( 1 + (0.00866 - 0.00866i)T - 13iT^{2} \)
17 \( 1 + 2.70T + 17T^{2} \)
19 \( 1 + (-0.000580 + 0.000580i)T - 19iT^{2} \)
23 \( 1 - 5.76iT - 23T^{2} \)
29 \( 1 + (4.44 - 4.44i)T - 29iT^{2} \)
31 \( 1 + 6.14T + 31T^{2} \)
37 \( 1 + (-0.123 - 0.123i)T + 37iT^{2} \)
41 \( 1 + 3.60iT - 41T^{2} \)
43 \( 1 + (-5.21 - 5.21i)T + 43iT^{2} \)
47 \( 1 + 9.30T + 47T^{2} \)
53 \( 1 + (6.50 + 6.50i)T + 53iT^{2} \)
59 \( 1 + (-8.31 - 8.31i)T + 59iT^{2} \)
61 \( 1 + (-4.66 + 4.66i)T - 61iT^{2} \)
67 \( 1 + (-4.31 + 4.31i)T - 67iT^{2} \)
71 \( 1 - 8.93iT - 71T^{2} \)
73 \( 1 + 6.63iT - 73T^{2} \)
79 \( 1 + 8.30T + 79T^{2} \)
83 \( 1 + (6.34 - 6.34i)T - 83iT^{2} \)
89 \( 1 + 6.68iT - 89T^{2} \)
97 \( 1 - 8.80T + 97T^{2} \)
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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

−11.66201917699106271235787326070, −10.95851756650165143816823176613, −9.662166962733559212782041180384, −8.461315147459432408673826555824, −7.47723822316729198776496632551, −6.75565178414871545577049020480, −5.55350869460516112038265328981, −4.34740823828750136247887696191, −3.35342131012643331737619130101, −1.83970120931991054368961957715, 2.18186140793384758675687874781, 3.39523707506008801975219518818, 4.38608978791495679513802018813, 5.51844870420647881280035580970, 6.60109126419609731222052083942, 7.65537979294688042379641238866, 8.832962006027260068130638846372, 9.882617254640820847020383686088, 10.97494777807184745195986340559, 11.48630208854060012964262398869

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