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Properties

Label 2-336-48.35-c1-0-34
Degree $2$
Conductor $336$
Sign $0.975 + 0.220i$
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 − i)2-s + (1.41 + i)3-s − 2i·4-s + (2.41 + 2.41i)5-s + (2.41 − 0.414i)6-s − 7-s + (−2 − 2i)8-s + (1.00 + 2.82i)9-s + 4.82·10-s + (−1.82 + 1.82i)11-s + (2 − 2.82i)12-s + (−1.58 − 1.58i)13-s + (−1 + i)14-s + (1 + 5.82i)15-s − 4·16-s − 6.82i·17-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.816 + 0.577i)3-s − i·4-s + (1.07 + 1.07i)5-s + (0.985 − 0.169i)6-s − 0.377·7-s + (−0.707 − 0.707i)8-s + (0.333 + 0.942i)9-s + 1.52·10-s + (−0.551 + 0.551i)11-s + (0.577 − 0.816i)12-s + (−0.439 − 0.439i)13-s + (−0.267 + 0.267i)14-s + (0.258 + 1.50i)15-s − 16-s − 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.975 + 0.220i$
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: \((2,\ 336,\ (\ :1/2),\ 0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.52349 - 0.282261i\)
\(L(\frac12)\) \(\approx\) \(2.52349 - 0.282261i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 + (-1.41 - i)T \)
7 \( 1 + T \)
good5 \( 1 + (-2.41 - 2.41i)T + 5iT^{2} \)
11 \( 1 + (1.82 - 1.82i)T - 11iT^{2} \)
13 \( 1 + (1.58 + 1.58i)T + 13iT^{2} \)
17 \( 1 + 6.82iT - 17T^{2} \)
19 \( 1 + (2.41 - 2.41i)T - 19iT^{2} \)
23 \( 1 + 3.65iT - 23T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 + 10.4iT - 31T^{2} \)
37 \( 1 + (3.82 - 3.82i)T - 37iT^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 + (-1.82 - 1.82i)T + 43iT^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + (-0.171 - 0.171i)T + 53iT^{2} \)
59 \( 1 + (4.07 - 4.07i)T - 59iT^{2} \)
61 \( 1 + (0.414 + 0.414i)T + 61iT^{2} \)
67 \( 1 + (7 - 7i)T - 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + (-10.8 - 10.8i)T + 83iT^{2} \)
89 \( 1 - 4.34T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−11.34551505700909568322201034416, −10.35693929258689751472294782662, −9.940739651968815769900181674535, −9.285611556342332866260999489538, −7.63280501647581216112370897276, −6.49391578654705675872044955266, −5.39640746288363285911023579244, −4.26256575707041920285728388327, −2.73925746450874589906940968989, −2.45204890383109105561336242009, 1.89418719046149637377031316571, 3.30363204555263239169880498292, 4.68929831830903392125508341146, 5.82246070762017210792379448642, 6.60003522144711863808979598495, 7.80146935661512625280145224448, 8.767734310781057890338096985306, 9.193072186789480298107268255907, 10.59268617310018083081074305567, 12.31651913229741636167001532121

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