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Properties

Label 2-336-48.35-c1-0-36
Degree $2$
Conductor $336$
Sign $-0.533 + 0.845i$
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 − i)2-s + (−1.41 + i)3-s − 2i·4-s + (−0.414 − 0.414i)5-s + (−0.414 + 2.41i)6-s − 7-s + (−2 − 2i)8-s + (1.00 − 2.82i)9-s − 0.828·10-s + (3.82 − 3.82i)11-s + (2 + 2.82i)12-s + (−4.41 − 4.41i)13-s + (−1 + i)14-s + (1 + 0.171i)15-s − 4·16-s − 1.17i·17-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.816 + 0.577i)3-s − i·4-s + (−0.185 − 0.185i)5-s + (−0.169 + 0.985i)6-s − 0.377·7-s + (−0.707 − 0.707i)8-s + (0.333 − 0.942i)9-s − 0.261·10-s + (1.15 − 1.15i)11-s + (0.577 + 0.816i)12-s + (−1.22 − 1.22i)13-s + (−0.267 + 0.267i)14-s + (0.258 + 0.0442i)15-s − 16-s − 0.284i·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.533 + 0.845i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.570464 - 1.03415i\)
\(L(\frac12)\) \(\approx\) \(0.570464 - 1.03415i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - i)T \)
7 \( 1 + T \)
good5 \( 1 + (0.414 + 0.414i)T + 5iT^{2} \)
11 \( 1 + (-3.82 + 3.82i)T - 11iT^{2} \)
13 \( 1 + (4.41 + 4.41i)T + 13iT^{2} \)
17 \( 1 + 1.17iT - 17T^{2} \)
19 \( 1 + (-0.414 + 0.414i)T - 19iT^{2} \)
23 \( 1 - 7.65iT - 23T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 - 6.48iT - 31T^{2} \)
37 \( 1 + (-1.82 + 1.82i)T - 37iT^{2} \)
41 \( 1 - 0.343T + 41T^{2} \)
43 \( 1 + (3.82 + 3.82i)T + 43iT^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + (-5.82 - 5.82i)T + 53iT^{2} \)
59 \( 1 + (-10.0 + 10.0i)T - 59iT^{2} \)
61 \( 1 + (-2.41 - 2.41i)T + 61iT^{2} \)
67 \( 1 + (7 - 7i)T - 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 - 5.17iT - 73T^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + (8.89 + 8.89i)T + 83iT^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
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

−11.49333552977705537644835032118, −10.39564968877218846229018859475, −9.789738136599274957711632491911, −8.793905522854758155912393604264, −7.05146736209596164404617231508, −5.93641762778639596123673914671, −5.22768031196217088281807351875, −4.03915799548784227572736964501, −3.06594872541961101916096390310, −0.74338045163897465537553786839, 2.23517625318762102262077562984, 4.14544776281565325000678378912, 4.89844430213316792687765918313, 6.30362131573968834640199601759, 6.87818205754557574638725320696, 7.53897129773797087448189182808, 8.953014273312896316789669847365, 10.02609051729489962925177899520, 11.43174642101361744536766838495, 12.08382290197867488857180290686

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