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Properties

Label 2-336-48.35-c1-0-12
Degree $2$
Conductor $336$
Sign $-0.479 - 0.877i$
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  + (0.786 + 1.17i)2-s + (1.34 + 1.09i)3-s + (−0.762 + 1.84i)4-s + (−1.07 − 1.07i)5-s + (−0.223 + 2.43i)6-s + 7-s + (−2.77 + 0.557i)8-s + (0.620 + 2.93i)9-s + (0.417 − 2.10i)10-s + (−2.21 + 2.21i)11-s + (−3.04 + 1.65i)12-s + (3.25 + 3.25i)13-s + (0.786 + 1.17i)14-s + (−0.273 − 2.61i)15-s + (−2.83 − 2.82i)16-s + 2.00i·17-s + ⋯
L(s)  = 1  + (0.556 + 0.831i)2-s + (0.776 + 0.629i)3-s + (−0.381 + 0.924i)4-s + (−0.480 − 0.480i)5-s + (−0.0914 + 0.995i)6-s + 0.377·7-s + (−0.980 + 0.197i)8-s + (0.206 + 0.978i)9-s + (0.132 − 0.666i)10-s + (−0.666 + 0.666i)11-s + (−0.878 + 0.477i)12-s + (0.902 + 0.902i)13-s + (0.210 + 0.314i)14-s + (−0.0705 − 0.675i)15-s + (−0.709 − 0.705i)16-s + 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.479 - 0.877i$
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: \((2,\ 336,\ (\ :1/2),\ -0.479 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01046 + 1.70272i\)
\(L(\frac12)\) \(\approx\) \(1.01046 + 1.70272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.786 - 1.17i)T \)
3 \( 1 + (-1.34 - 1.09i)T \)
7 \( 1 - T \)
good5 \( 1 + (1.07 + 1.07i)T + 5iT^{2} \)
11 \( 1 + (2.21 - 2.21i)T - 11iT^{2} \)
13 \( 1 + (-3.25 - 3.25i)T + 13iT^{2} \)
17 \( 1 - 2.00iT - 17T^{2} \)
19 \( 1 + (-2.74 + 2.74i)T - 19iT^{2} \)
23 \( 1 + 3.98iT - 23T^{2} \)
29 \( 1 + (-3.66 + 3.66i)T - 29iT^{2} \)
31 \( 1 + 7.29iT - 31T^{2} \)
37 \( 1 + (-4.74 + 4.74i)T - 37iT^{2} \)
41 \( 1 - 2.69T + 41T^{2} \)
43 \( 1 + (-1.69 - 1.69i)T + 43iT^{2} \)
47 \( 1 + 0.231T + 47T^{2} \)
53 \( 1 + (5.54 + 5.54i)T + 53iT^{2} \)
59 \( 1 + (10.1 - 10.1i)T - 59iT^{2} \)
61 \( 1 + (-10.2 - 10.2i)T + 61iT^{2} \)
67 \( 1 + (10.6 - 10.6i)T - 67iT^{2} \)
71 \( 1 - 1.41iT - 71T^{2} \)
73 \( 1 + 5.10iT - 73T^{2} \)
79 \( 1 - 0.216iT - 79T^{2} \)
83 \( 1 + (3.49 + 3.49i)T + 83iT^{2} \)
89 \( 1 - 1.48T + 89T^{2} \)
97 \( 1 + 5.51T + 97T^{2} \)
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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

−12.00750273033455696398625688738, −11.01832587212576709586183484978, −9.721068812499732399582982051010, −8.759659225408657953884302183340, −8.117913091545641893315180514324, −7.27865098665162079206358166014, −5.88438812867901583045528335742, −4.51359923469228638047157440485, −4.19864037037731890688971578783, −2.59123616527918470749095594605, 1.26992575397022011513649442138, 2.99910691262644090786714993497, 3.49673063480595738855535639606, 5.15852201813217449321642243994, 6.28595901720232714089199594445, 7.60051352977959402936346135384, 8.390242406722138610000798637008, 9.462915740291760408243329344278, 10.61148582909407681392867943660, 11.28829790615870881445171559438

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