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Properties

Label 2-336-48.35-c1-0-0
Degree $2$
Conductor $336$
Sign $0.660 - 0.750i$
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  + (−0.244 − 1.39i)2-s + (−1.68 − 0.408i)3-s + (−1.88 + 0.679i)4-s + (−2.26 − 2.26i)5-s + (−0.157 + 2.44i)6-s − 7-s + (1.40 + 2.45i)8-s + (2.66 + 1.37i)9-s + (−2.59 + 3.70i)10-s + (−1.82 + 1.82i)11-s + (3.44 − 0.377i)12-s + (0.915 + 0.915i)13-s + (0.244 + 1.39i)14-s + (2.88 + 4.72i)15-s + (3.07 − 2.55i)16-s − 2.93i·17-s + ⋯
L(s)  = 1  + (−0.172 − 0.984i)2-s + (−0.971 − 0.235i)3-s + (−0.940 + 0.339i)4-s + (−1.01 − 1.01i)5-s + (−0.0643 + 0.997i)6-s − 0.377·7-s + (0.497 + 0.867i)8-s + (0.889 + 0.457i)9-s + (−0.821 + 1.17i)10-s + (−0.548 + 0.548i)11-s + (0.994 − 0.108i)12-s + (0.253 + 0.253i)13-s + (0.0652 + 0.372i)14-s + (0.744 + 1.22i)15-s + (0.768 − 0.639i)16-s − 0.711i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.660 - 0.750i$
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: \((2,\ 336,\ (\ :1/2),\ 0.660 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.161231 + 0.0728410i\)
\(L(\frac12)\) \(\approx\) \(0.161231 + 0.0728410i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.244 + 1.39i)T \)
3 \( 1 + (1.68 + 0.408i)T \)
7 \( 1 + T \)
good5 \( 1 + (2.26 + 2.26i)T + 5iT^{2} \)
11 \( 1 + (1.82 - 1.82i)T - 11iT^{2} \)
13 \( 1 + (-0.915 - 0.915i)T + 13iT^{2} \)
17 \( 1 + 2.93iT - 17T^{2} \)
19 \( 1 + (-1.79 + 1.79i)T - 19iT^{2} \)
23 \( 1 - 8.77iT - 23T^{2} \)
29 \( 1 + (1.67 - 1.67i)T - 29iT^{2} \)
31 \( 1 - 8.15iT - 31T^{2} \)
37 \( 1 + (4.29 - 4.29i)T - 37iT^{2} \)
41 \( 1 - 4.52T + 41T^{2} \)
43 \( 1 + (3.46 + 3.46i)T + 43iT^{2} \)
47 \( 1 + 8.60T + 47T^{2} \)
53 \( 1 + (6.48 + 6.48i)T + 53iT^{2} \)
59 \( 1 + (8.08 - 8.08i)T - 59iT^{2} \)
61 \( 1 + (-2.11 - 2.11i)T + 61iT^{2} \)
67 \( 1 + (2.17 - 2.17i)T - 67iT^{2} \)
71 \( 1 + 4.72iT - 71T^{2} \)
73 \( 1 + 8.88iT - 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + (-7.45 - 7.45i)T + 83iT^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 - 9.08T + 97T^{2} \)
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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

−11.75509342493773625556435049130, −11.03935458833832306440482431278, −9.945103519132092358212404153071, −9.101926565666539064348107401309, −7.954948805227659497583131233443, −7.10337825511057794141306999139, −5.29177063851893132380929604103, −4.70896922227579515934776669102, −3.44456285847143785748905648644, −1.39071428963365009549665111463, 0.16331840498372903654157228430, 3.47181194210714422472702769959, 4.49164844038484439163040633982, 5.86811659993072907739187192821, 6.48401416307773158820884509931, 7.50736547434016028944946070690, 8.278403452290793770535221936296, 9.686385200704119464797734842562, 10.60398327097501884980909785252, 11.13710702423154598900587525377

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