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Properties

Label 2-336-21.5-c3-0-6
Degree $2$
Conductor $336$
Sign $-0.379 - 0.925i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.99 + 1.42i)3-s + (−6.11 + 10.5i)5-s + (−4.39 − 17.9i)7-s + (22.9 − 14.1i)9-s + (47.3 − 27.3i)11-s + 27.0i·13-s + (15.5 − 61.5i)15-s + (20.3 + 35.1i)17-s + (48.4 + 27.9i)19-s + (47.5 + 83.6i)21-s + (−93.3 − 53.8i)23-s + (−12.1 − 21.1i)25-s + (−94.6 + 103. i)27-s − 38.2i·29-s + (−257. + 148. i)31-s + ⋯
L(s)  = 1  + (−0.961 + 0.273i)3-s + (−0.546 + 0.946i)5-s + (−0.237 − 0.971i)7-s + (0.850 − 0.525i)9-s + (1.29 − 0.749i)11-s + 0.576i·13-s + (0.267 − 1.06i)15-s + (0.289 + 0.501i)17-s + (0.585 + 0.338i)19-s + (0.493 + 0.869i)21-s + (−0.846 − 0.488i)23-s + (−0.0975 − 0.168i)25-s + (−0.674 + 0.738i)27-s − 0.244i·29-s + (−1.49 + 0.862i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.379 - 0.925i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.379 - 0.925i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8194203934\)
\(L(\frac12)\) \(\approx\) \(0.8194203934\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (4.99 - 1.42i)T \)
7 \( 1 + (4.39 + 17.9i)T \)
good5 \( 1 + (6.11 - 10.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-47.3 + 27.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 27.0iT - 2.19e3T^{2} \)
17 \( 1 + (-20.3 - 35.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-48.4 - 27.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (93.3 + 53.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 38.2iT - 2.43e4T^{2} \)
31 \( 1 + (257. - 148. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (142. - 246. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 28.3T + 6.89e4T^{2} \)
43 \( 1 + 212.T + 7.95e4T^{2} \)
47 \( 1 + (-125. + 216. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-294. + 170. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-451. - 781. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-499. - 288. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-184. - 319. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.18e3iT - 3.57e5T^{2} \)
73 \( 1 + (407. - 235. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-20.0 + 34.7i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 855.T + 5.71e5T^{2} \)
89 \( 1 + (-322. + 557. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 748. iT - 9.12e5T^{2} \)
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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

−11.50975305135545525534380295628, −10.52230535011896542363869631006, −9.950809544104263582159972895112, −8.633597378495703627763283893983, −7.14750649834527461792980078561, −6.75447083937999299841601081664, −5.66515219411712006716337711710, −4.05928148716641793682905448622, −3.57204081662857263628323940862, −1.19824268117025255824208294698, 0.38852478062741005035470470920, 1.80623259439063768195365142713, 3.82864128205149921825897519898, 5.00673323374131072619575566003, 5.76120957573235474554177333757, 6.93213473651859401549260022945, 7.903220684693957206642591841833, 9.112017105100212459569449501718, 9.749601655455529897420487867884, 11.16212835590427692225115125604

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