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Properties

Label 2-336-48.11-c1-0-26
Degree $2$
Conductor $336$
Sign $0.943 - 0.331i$
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  + (1.30 − 0.533i)2-s + (1.18 + 1.26i)3-s + (1.42 − 1.39i)4-s + (−1.97 + 1.97i)5-s + (2.22 + 1.02i)6-s + 7-s + (1.12 − 2.59i)8-s + (−0.197 + 2.99i)9-s + (−1.53 + 3.64i)10-s + (0.735 + 0.735i)11-s + (3.46 + 0.152i)12-s + (2.83 − 2.83i)13-s + (1.30 − 0.533i)14-s + (−4.84 − 0.159i)15-s + (0.0887 − 3.99i)16-s + 3.89i·17-s + ⋯
L(s)  = 1  + (0.925 − 0.377i)2-s + (0.683 + 0.730i)3-s + (0.714 − 0.699i)4-s + (−0.885 + 0.885i)5-s + (0.908 + 0.418i)6-s + 0.377·7-s + (0.397 − 0.917i)8-s + (−0.0659 + 0.997i)9-s + (−0.485 + 1.15i)10-s + (0.221 + 0.221i)11-s + (0.999 + 0.0440i)12-s + (0.787 − 0.787i)13-s + (0.349 − 0.142i)14-s + (−1.25 − 0.0413i)15-s + (0.0221 − 0.999i)16-s + 0.944i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47960 + 0.422471i\)
\(L(\frac12)\) \(\approx\) \(2.47960 + 0.422471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.533i)T \)
3 \( 1 + (-1.18 - 1.26i)T \)
7 \( 1 - T \)
good5 \( 1 + (1.97 - 1.97i)T - 5iT^{2} \)
11 \( 1 + (-0.735 - 0.735i)T + 11iT^{2} \)
13 \( 1 + (-2.83 + 2.83i)T - 13iT^{2} \)
17 \( 1 - 3.89iT - 17T^{2} \)
19 \( 1 + (2.65 + 2.65i)T + 19iT^{2} \)
23 \( 1 + 4.59iT - 23T^{2} \)
29 \( 1 + (6.93 + 6.93i)T + 29iT^{2} \)
31 \( 1 - 6.03iT - 31T^{2} \)
37 \( 1 + (7.23 + 7.23i)T + 37iT^{2} \)
41 \( 1 - 0.424T + 41T^{2} \)
43 \( 1 + (1.72 - 1.72i)T - 43iT^{2} \)
47 \( 1 - 3.45T + 47T^{2} \)
53 \( 1 + (-7.07 + 7.07i)T - 53iT^{2} \)
59 \( 1 + (1.95 + 1.95i)T + 59iT^{2} \)
61 \( 1 + (0.0653 - 0.0653i)T - 61iT^{2} \)
67 \( 1 + (-9.77 - 9.77i)T + 67iT^{2} \)
71 \( 1 - 8.50iT - 71T^{2} \)
73 \( 1 + 5.13iT - 73T^{2} \)
79 \( 1 - 2.65iT - 79T^{2} \)
83 \( 1 + (0.666 - 0.666i)T - 83iT^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−11.36913030040617921786055118930, −10.80193412577809020830913072561, −10.20993953470084666822032633205, −8.768455327010125971049996655150, −7.79017101219562122396805240035, −6.73192815875406437692053629673, −5.42314187185107275762004309635, −4.11004103232454555600966817108, −3.56346805753261544631099524714, −2.29767050768685035823699872710, 1.69021815745708716889236269594, 3.44299980655306588934359895770, 4.27013481095600634839634366713, 5.55549817208795100544458187670, 6.79451223719347597506380736778, 7.65189220109743902174841136032, 8.438697779147953836458071806656, 9.168126329885759163733926794601, 11.11756398302927113580127953672, 11.86750498353288885106245267741

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