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Properties

Label 2-336-48.35-c1-0-46
Degree $2$
Conductor $336$
Sign $-0.780 - 0.624i$
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  + (−0.277 − 1.38i)2-s + (0.490 − 1.66i)3-s + (−1.84 + 0.770i)4-s + (−2.27 − 2.27i)5-s + (−2.43 − 0.218i)6-s + 7-s + (1.58 + 2.34i)8-s + (−2.51 − 1.62i)9-s + (−2.51 + 3.77i)10-s + (1.28 − 1.28i)11-s + (0.374 + 3.44i)12-s + (−2.15 − 2.15i)13-s + (−0.277 − 1.38i)14-s + (−4.88 + 2.65i)15-s + (2.81 − 2.84i)16-s + 6.50i·17-s + ⋯
L(s)  = 1  + (−0.196 − 0.980i)2-s + (0.283 − 0.959i)3-s + (−0.922 + 0.385i)4-s + (−1.01 − 1.01i)5-s + (−0.996 − 0.0893i)6-s + 0.377·7-s + (0.558 + 0.829i)8-s + (−0.839 − 0.543i)9-s + (−0.796 + 1.19i)10-s + (0.388 − 0.388i)11-s + (0.107 + 0.994i)12-s + (−0.597 − 0.597i)13-s + (−0.0742 − 0.370i)14-s + (−1.26 + 0.686i)15-s + (0.703 − 0.710i)16-s + 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.780 - 0.624i$
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: \((2,\ 336,\ (\ :1/2),\ -0.780 - 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.251187 + 0.716200i\)
\(L(\frac12)\) \(\approx\) \(0.251187 + 0.716200i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.277 + 1.38i)T \)
3 \( 1 + (-0.490 + 1.66i)T \)
7 \( 1 - T \)
good5 \( 1 + (2.27 + 2.27i)T + 5iT^{2} \)
11 \( 1 + (-1.28 + 1.28i)T - 11iT^{2} \)
13 \( 1 + (2.15 + 2.15i)T + 13iT^{2} \)
17 \( 1 - 6.50iT - 17T^{2} \)
19 \( 1 + (3.42 - 3.42i)T - 19iT^{2} \)
23 \( 1 + 5.60iT - 23T^{2} \)
29 \( 1 + (-3.59 + 3.59i)T - 29iT^{2} \)
31 \( 1 - 0.730iT - 31T^{2} \)
37 \( 1 + (-7.94 + 7.94i)T - 37iT^{2} \)
41 \( 1 - 3.23T + 41T^{2} \)
43 \( 1 + (8.55 + 8.55i)T + 43iT^{2} \)
47 \( 1 + 7.39T + 47T^{2} \)
53 \( 1 + (-0.785 - 0.785i)T + 53iT^{2} \)
59 \( 1 + (-4.42 + 4.42i)T - 59iT^{2} \)
61 \( 1 + (1.23 + 1.23i)T + 61iT^{2} \)
67 \( 1 + (2.62 - 2.62i)T - 67iT^{2} \)
71 \( 1 - 1.31iT - 71T^{2} \)
73 \( 1 + 10.4iT - 73T^{2} \)
79 \( 1 + 9.74iT - 79T^{2} \)
83 \( 1 + (0.603 + 0.603i)T + 83iT^{2} \)
89 \( 1 - 0.657T + 89T^{2} \)
97 \( 1 - 8.66T + 97T^{2} \)
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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

−11.23393858223622966512143023868, −10.21189605535751219121506267982, −8.752330602789216114478485470579, −8.349083091293604556071326285393, −7.69901303886104828201153261214, −6.03609224067043418538177436702, −4.58069861330712324251708828318, −3.60726063953617187652437451456, −2.00148737322370296002116259911, −0.56266955121126347978366671075, 3.00943449261720301221658856908, 4.31776055526361086827506875950, 4.97442678992851118313735793134, 6.60728398113868453348947851386, 7.37045040250665538265360185822, 8.247360222647402458226753294616, 9.336156679138979241065776273209, 9.979198258172790711089969417457, 11.20981047369359564983335274371, 11.68960030691943004519845473415

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