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Properties

Label 2-336-48.11-c1-0-19
Degree $2$
Conductor $336$
Sign $-0.845 + 0.533i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−1 + 1.41i)3-s + 2i·4-s + (−2.41 + 2.41i)5-s + (2.41 − 0.414i)6-s − 7-s + (2 − 2i)8-s + (−1.00 − 2.82i)9-s + 4.82·10-s + (1.82 + 1.82i)11-s + (−2.82 − 2i)12-s + (−1.58 + 1.58i)13-s + (1 + i)14-s + (−1 − 5.82i)15-s − 4·16-s − 6.82i·17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.577 + 0.816i)3-s + i·4-s + (−1.07 + 1.07i)5-s + (0.985 − 0.169i)6-s − 0.377·7-s + (0.707 − 0.707i)8-s + (−0.333 − 0.942i)9-s + 1.52·10-s + (0.551 + 0.551i)11-s + (−0.816 − 0.577i)12-s + (−0.439 + 0.439i)13-s + (0.267 + 0.267i)14-s + (−0.258 − 1.50i)15-s − 16-s − 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.845 + 0.533i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.845 + 0.533i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (1 - 1.41i)T \)
7 \( 1 + T \)
good5 \( 1 + (2.41 - 2.41i)T - 5iT^{2} \)
11 \( 1 + (-1.82 - 1.82i)T + 11iT^{2} \)
13 \( 1 + (1.58 - 1.58i)T - 13iT^{2} \)
17 \( 1 + 6.82iT - 17T^{2} \)
19 \( 1 + (2.41 + 2.41i)T + 19iT^{2} \)
23 \( 1 + 3.65iT - 23T^{2} \)
29 \( 1 + (3 + 3i)T + 29iT^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 + (3.82 + 3.82i)T + 37iT^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + (-1.82 + 1.82i)T - 43iT^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + (0.171 - 0.171i)T - 53iT^{2} \)
59 \( 1 + (-4.07 - 4.07i)T + 59iT^{2} \)
61 \( 1 + (0.414 - 0.414i)T - 61iT^{2} \)
67 \( 1 + (7 + 7i)T + 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (10.8 - 10.8i)T - 83iT^{2} \)
89 \( 1 + 4.34T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−11.05552938994961423218347413765, −10.41221129766235444486958744367, −9.519031187647198326560212170575, −8.687419532715339279250843899999, −7.11917160694813217799537031794, −6.84336716455709595735069534256, −4.75449862668910935310008206388, −3.78105786482706271818575895923, −2.77563324287679678633895252305, 0, 1.48238546821133835778694109709, 4.03931365284095312958045789388, 5.39539661036086140239566106526, 6.20035190925488296874818208898, 7.33355047843592947998284616037, 8.182201244163119704056554879272, 8.675153159905999938793793119879, 10.01453382017577529359200937636, 11.08302588025099201369877890334

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