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Properties

Label 2-336-21.5-c3-0-35
Degree $2$
Conductor $336$
Sign $0.0428 + 0.999i$
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  + (4.27 − 2.94i)3-s + (−5.57 + 9.65i)5-s + (7.78 − 16.8i)7-s + (9.60 − 25.2i)9-s + (0.436 − 0.251i)11-s − 15.9i·13-s + (4.62 + 57.7i)15-s + (−41.0 − 71.0i)17-s + (−55.9 − 32.2i)19-s + (−16.2 − 94.8i)21-s + (159. + 92.0i)23-s + (0.337 + 0.583i)25-s + (−33.2 − 136. i)27-s − 289. i·29-s + (125. − 72.5i)31-s + ⋯
L(s)  = 1  + (0.823 − 0.567i)3-s + (−0.498 + 0.863i)5-s + (0.420 − 0.907i)7-s + (0.355 − 0.934i)9-s + (0.0119 − 0.00690i)11-s − 0.339i·13-s + (0.0795 + 0.994i)15-s + (−0.585 − 1.01i)17-s + (−0.675 − 0.389i)19-s + (−0.168 − 0.985i)21-s + (1.44 + 0.834i)23-s + (0.00269 + 0.00467i)25-s + (−0.237 − 0.971i)27-s − 1.85i·29-s + (0.727 − 0.420i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0428 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0428 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.0428 + 0.999i$
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: \((2,\ 336,\ (\ :3/2),\ 0.0428 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.115931918\)
\(L(\frac12)\) \(\approx\) \(2.115931918\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.27 + 2.94i)T \)
7 \( 1 + (-7.78 + 16.8i)T \)
good5 \( 1 + (5.57 - 9.65i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-0.436 + 0.251i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 15.9iT - 2.19e3T^{2} \)
17 \( 1 + (41.0 + 71.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (55.9 + 32.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-159. - 92.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 289. iT - 2.43e4T^{2} \)
31 \( 1 + (-125. + 72.5i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-101. + 176. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 52.6T + 6.89e4T^{2} \)
43 \( 1 + 111.T + 7.95e4T^{2} \)
47 \( 1 + (-126. + 218. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (588. - 339. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (249. + 432. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-222. - 128. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-44.0 - 76.3i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 717. iT - 3.57e5T^{2} \)
73 \( 1 + (303. - 175. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-480. + 832. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 177.T + 5.71e5T^{2} \)
89 \( 1 + (38.5 - 66.7i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 400. iT - 9.12e5T^{2} \)
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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

−11.07623281268985382803723170958, −9.939600198620422454842844921671, −8.941962717823443348243206192009, −7.76214999790762251100908868892, −7.30374378118305951594471497860, −6.40449794526747587171146952321, −4.60040928609690330430613599399, −3.47834639753567322228181311796, −2.41817154875968952240044934560, −0.70331895909165142124581662030, 1.62821168941421981820361003523, 3.01573624624700796172869546644, 4.41428885730853590087549303406, 5.00763439410398684534497992706, 6.53470536689456830265279066445, 8.009447280529930863114421602817, 8.674527847905017272065050108843, 9.087077361422434507098729133253, 10.43379147147055774557319022673, 11.23997129915606962530632236073

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