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Properties

Label 2-336-12.11-c3-0-8
Degree $2$
Conductor $336$
Sign $-0.128 - 0.991i$
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  + (−1.99 + 4.79i)3-s − 16.7i·5-s + 7i·7-s + (−19.0 − 19.1i)9-s − 60.9·11-s + 73.9·13-s + (80.2 + 33.4i)15-s + 16.3i·17-s + 153. i·19-s + (−33.5 − 13.9i)21-s + 111.·23-s − 154.·25-s + (129. − 52.8i)27-s + 155. i·29-s − 53.7i·31-s + ⋯
L(s)  = 1  + (−0.384 + 0.923i)3-s − 1.49i·5-s + 0.377i·7-s + (−0.704 − 0.709i)9-s − 1.66·11-s + 1.57·13-s + (1.38 + 0.575i)15-s + 0.232i·17-s + 1.85i·19-s + (−0.348 − 0.145i)21-s + 1.01·23-s − 1.23·25-s + (0.926 − 0.377i)27-s + 0.993i·29-s − 0.311i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.128 - 0.991i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.128 - 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.073759746\)
\(L(\frac12)\) \(\approx\) \(1.073759746\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.99 - 4.79i)T \)
7 \( 1 - 7iT \)
good5 \( 1 + 16.7iT - 125T^{2} \)
11 \( 1 + 60.9T + 1.33e3T^{2} \)
13 \( 1 - 73.9T + 2.19e3T^{2} \)
17 \( 1 - 16.3iT - 4.91e3T^{2} \)
19 \( 1 - 153. iT - 6.85e3T^{2} \)
23 \( 1 - 111.T + 1.21e4T^{2} \)
29 \( 1 - 155. iT - 2.43e4T^{2} \)
31 \( 1 + 53.7iT - 2.97e4T^{2} \)
37 \( 1 + 63.0T + 5.06e4T^{2} \)
41 \( 1 - 458. iT - 6.89e4T^{2} \)
43 \( 1 - 95.5iT - 7.95e4T^{2} \)
47 \( 1 - 70.5T + 1.03e5T^{2} \)
53 \( 1 - 417. iT - 1.48e5T^{2} \)
59 \( 1 + 328.T + 2.05e5T^{2} \)
61 \( 1 + 637.T + 2.26e5T^{2} \)
67 \( 1 - 201. iT - 3.00e5T^{2} \)
71 \( 1 - 394.T + 3.57e5T^{2} \)
73 \( 1 - 977.T + 3.89e5T^{2} \)
79 \( 1 - 514. iT - 4.93e5T^{2} \)
83 \( 1 + 1.13e3T + 5.71e5T^{2} \)
89 \( 1 - 1.03e3iT - 7.04e5T^{2} \)
97 \( 1 - 995.T + 9.12e5T^{2} \)
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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

−11.21193630042033726401318465785, −10.49051703986459825592547525314, −9.489065121843147919523112912322, −8.579317421568725306195314592606, −8.042780666518783512955458529646, −6.02528308712068596479665950941, −5.39168268848840231826560207816, −4.51022094561873314079819925626, −3.27661959543560545478527983512, −1.23494645533952699338259123800, 0.44301023599359910023731042814, 2.33679529727757028793335732872, 3.24595327203791481469886041074, 5.09965199009407643328555643924, 6.23553511008408159239283735026, 7.02900543398092708044409300143, 7.69194540088880058902341483674, 8.825246771799668809379758421970, 10.47478877388469959987911793145, 10.88270440521337133606490990986

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