Properties

Label 2-336-112.101-c2-0-32
Degree $2$
Conductor $336$
Sign $0.554 + 0.832i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0542 − 1.99i)2-s + (1.67 − 0.448i)3-s + (−3.99 − 0.216i)4-s + (4.85 + 1.30i)5-s + (−0.805 − 3.36i)6-s + (6.58 + 2.37i)7-s + (−0.650 + 7.97i)8-s + (2.59 − 1.50i)9-s + (2.86 − 9.64i)10-s + (−2.23 + 0.597i)11-s + (−6.77 + 1.42i)12-s + (6.27 + 6.27i)13-s + (5.11 − 13.0i)14-s + 8.71·15-s + (15.9 + 1.73i)16-s + (9.91 + 5.72i)17-s + ⋯
L(s)  = 1  + (0.0271 − 0.999i)2-s + (0.557 − 0.149i)3-s + (−0.998 − 0.0542i)4-s + (0.971 + 0.260i)5-s + (−0.134 − 0.561i)6-s + (0.940 + 0.339i)7-s + (−0.0813 + 0.996i)8-s + (0.288 − 0.166i)9-s + (0.286 − 0.964i)10-s + (−0.202 + 0.0543i)11-s + (−0.564 + 0.118i)12-s + (0.483 + 0.483i)13-s + (0.365 − 0.930i)14-s + 0.580·15-s + (0.994 + 0.108i)16-s + (0.583 + 0.336i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.554 + 0.832i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.554 + 0.832i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.10346 - 1.12600i\)
\(L(\frac12)\) \(\approx\) \(2.10346 - 1.12600i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0542 + 1.99i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 + (-6.58 - 2.37i)T \)
good5 \( 1 + (-4.85 - 1.30i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (2.23 - 0.597i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-6.27 - 6.27i)T + 169iT^{2} \)
17 \( 1 + (-9.91 - 5.72i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.98 - 14.8i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-28.0 + 16.2i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-1.14 + 1.14i)T - 841iT^{2} \)
31 \( 1 + (33.7 + 19.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-12.0 + 45.0i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 11.9T + 1.68e3T^{2} \)
43 \( 1 + (35.5 + 35.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (44.5 - 25.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (7.53 - 2.02i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-0.304 - 1.13i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (16.6 - 62.1i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (10.6 + 39.8i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 23.9iT - 5.04e3T^{2} \)
73 \( 1 + (-46.0 + 79.8i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (17.3 + 30.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-4.74 - 4.74i)T + 6.88e3iT^{2} \)
89 \( 1 + (-71.8 - 124. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 82.7iT - 9.40e3T^{2} \)
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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

−11.06626944033887413946876998403, −10.36795398272129180258523407454, −9.377191649021276407587063427203, −8.657708806332496174833183650401, −7.69276963781120125437537429168, −6.08929038794374218098681708035, −5.06014745453511728581479834696, −3.75563376987774486782822177188, −2.37842575821616511681447472906, −1.53193299915385401647430327394, 1.36611102445363446622637809472, 3.30468936244990726233587567109, 4.83449993146507246103153615966, 5.42676864680730308639615377506, 6.74749510332279016928963986356, 7.74914283492139995079316012194, 8.573365242024063266302107300108, 9.398053055657265645403999446929, 10.23972830063548964245760910824, 11.36229212475381436964948564392

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