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Properties

Label 2-336-48.11-c1-0-12
Degree $2$
Conductor $336$
Sign $0.285 - 0.958i$
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  + (−1.38 − 0.302i)2-s + (1.42 + 0.985i)3-s + (1.81 + 0.836i)4-s + (−0.766 + 0.766i)5-s + (−1.66 − 1.79i)6-s − 7-s + (−2.25 − 1.70i)8-s + (1.05 + 2.80i)9-s + (1.29 − 0.826i)10-s + (3.09 + 3.09i)11-s + (1.76 + 2.98i)12-s + (0.262 − 0.262i)13-s + (1.38 + 0.302i)14-s + (−1.84 + 0.336i)15-s + (2.60 + 3.03i)16-s + 2.68i·17-s + ⋯
L(s)  = 1  + (−0.976 − 0.213i)2-s + (0.822 + 0.568i)3-s + (0.908 + 0.418i)4-s + (−0.342 + 0.342i)5-s + (−0.681 − 0.731i)6-s − 0.377·7-s + (−0.797 − 0.602i)8-s + (0.353 + 0.935i)9-s + (0.408 − 0.261i)10-s + (0.932 + 0.932i)11-s + (0.509 + 0.860i)12-s + (0.0728 − 0.0728i)13-s + (0.369 + 0.0808i)14-s + (−0.476 + 0.0869i)15-s + (0.650 + 0.759i)16-s + 0.651i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.285 - 0.958i$
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: \((2,\ 336,\ (\ :1/2),\ 0.285 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823431 + 0.613876i\)
\(L(\frac12)\) \(\approx\) \(0.823431 + 0.613876i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.302i)T \)
3 \( 1 + (-1.42 - 0.985i)T \)
7 \( 1 + T \)
good5 \( 1 + (0.766 - 0.766i)T - 5iT^{2} \)
11 \( 1 + (-3.09 - 3.09i)T + 11iT^{2} \)
13 \( 1 + (-0.262 + 0.262i)T - 13iT^{2} \)
17 \( 1 - 2.68iT - 17T^{2} \)
19 \( 1 + (2.44 + 2.44i)T + 19iT^{2} \)
23 \( 1 - 3.24iT - 23T^{2} \)
29 \( 1 + (3.28 + 3.28i)T + 29iT^{2} \)
31 \( 1 + 3.76iT - 31T^{2} \)
37 \( 1 + (-4.88 - 4.88i)T + 37iT^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 + (0.938 - 0.938i)T - 43iT^{2} \)
47 \( 1 - 0.764T + 47T^{2} \)
53 \( 1 + (4.14 - 4.14i)T - 53iT^{2} \)
59 \( 1 + (2.63 + 2.63i)T + 59iT^{2} \)
61 \( 1 + (-10.7 + 10.7i)T - 61iT^{2} \)
67 \( 1 + (8.41 + 8.41i)T + 67iT^{2} \)
71 \( 1 + 13.7iT - 71T^{2} \)
73 \( 1 + 6.79iT - 73T^{2} \)
79 \( 1 - 3.83iT - 79T^{2} \)
83 \( 1 + (0.107 - 0.107i)T - 83iT^{2} \)
89 \( 1 - 9.00T + 89T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−11.38890365985832486163515674140, −10.69161694296072233107742065045, −9.510313989717030508215842969655, −9.335179815892173339781485959480, −8.041904571704418562516811955530, −7.34139944464968760286235252009, −6.24090646896017280376146414857, −4.28458457615491714534257189109, −3.29532020408953495635160309919, −1.95630231070154892763446203994, 0.948529813585664423775363845043, 2.58213458562632058704732746582, 3.92127048543160830041156557367, 5.94043731206179282299195532797, 6.78565958156057625804324475386, 7.74055126938447124976269304538, 8.704156781609282468722320203846, 9.092250728700165666256100122004, 10.20619059355095788350498115938, 11.35451402960122927838279096281

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