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Properties

Label 2-336-336.5-c1-0-11
Degree $2$
Conductor $336$
Sign $-0.393 - 0.919i$
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  + (−1.35 + 0.415i)2-s + (1.14 + 1.30i)3-s + (1.65 − 1.12i)4-s + (0.248 − 0.0665i)5-s + (−2.08 − 1.28i)6-s + (0.154 + 2.64i)7-s + (−1.77 + 2.20i)8-s + (−0.390 + 2.97i)9-s + (−0.308 + 0.193i)10-s + (−5.69 − 1.52i)11-s + (3.35 + 0.872i)12-s + (1.65 + 1.65i)13-s + (−1.30 − 3.50i)14-s + (0.370 + 0.247i)15-s + (1.47 − 3.71i)16-s + (3.31 + 5.73i)17-s + ⋯
L(s)  = 1  + (−0.955 + 0.293i)2-s + (0.659 + 0.751i)3-s + (0.827 − 0.561i)4-s + (0.111 − 0.0297i)5-s + (−0.851 − 0.524i)6-s + (0.0584 + 0.998i)7-s + (−0.626 + 0.779i)8-s + (−0.130 + 0.991i)9-s + (−0.0974 + 0.0610i)10-s + (−1.71 − 0.460i)11-s + (0.967 + 0.251i)12-s + (0.458 + 0.458i)13-s + (−0.349 − 0.937i)14-s + (0.0955 + 0.0638i)15-s + (0.369 − 0.929i)16-s + (0.803 + 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.393 - 0.919i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.393 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.541948 + 0.821185i\)
\(L(\frac12)\) \(\approx\) \(0.541948 + 0.821185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.415i)T \)
3 \( 1 + (-1.14 - 1.30i)T \)
7 \( 1 + (-0.154 - 2.64i)T \)
good5 \( 1 + (-0.248 + 0.0665i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (5.69 + 1.52i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.65 - 1.65i)T + 13iT^{2} \)
17 \( 1 + (-3.31 - 5.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.20 + 1.12i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.18 + 3.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.853 - 0.853i)T + 29iT^{2} \)
31 \( 1 + (3.82 - 2.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.93 + 1.05i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.05iT - 41T^{2} \)
43 \( 1 + (2.94 + 2.94i)T + 43iT^{2} \)
47 \( 1 + (0.356 - 0.618i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.8 - 3.18i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.338 - 1.26i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-14.1 + 3.80i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (6.87 + 1.84i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.22T + 71T^{2} \)
73 \( 1 + (2.87 + 4.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.19 + 3.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.70 + 6.70i)T - 83iT^{2} \)
89 \( 1 + (0.982 + 0.567i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.1iT - 97T^{2} \)
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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

−11.44428915251183791736581003456, −10.60995426771955550592185421909, −9.925131876814290834116540887719, −8.902723916872249223864704408517, −8.339337781305102025343492242233, −7.51062185027477989602293060381, −5.86039195951706466020410448461, −5.21914652365691784915432123764, −3.26783945281202899272272627109, −2.12947516735570403776344010705, 0.880685930857990999751836718826, 2.49345452322508822098873816835, 3.52060351103040038288978978321, 5.50187665434605319259742747955, 7.03636348361859470942438968699, 7.64889145936101326759005243362, 8.157460342366645371133213604156, 9.623411060993465698564431891413, 10.03214436236931686165876789383, 11.19489058047426633828036357312

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