Properties

Label 2-336-48.35-c1-0-22
Degree $2$
Conductor $336$
Sign $0.810 + 0.586i$
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  + (0.103 − 1.41i)2-s + (1.29 + 1.15i)3-s + (−1.97 − 0.291i)4-s + (0.186 + 0.186i)5-s + (1.75 − 1.70i)6-s + 7-s + (−0.615 + 2.76i)8-s + (0.350 + 2.97i)9-s + (0.282 − 0.243i)10-s + (4.29 − 4.29i)11-s + (−2.22 − 2.65i)12-s + (2.73 + 2.73i)13-s + (0.103 − 1.41i)14-s + (0.0267 + 0.455i)15-s + (3.83 + 1.15i)16-s − 4.98i·17-s + ⋯
L(s)  = 1  + (0.0730 − 0.997i)2-s + (0.747 + 0.664i)3-s + (−0.989 − 0.145i)4-s + (0.0833 + 0.0833i)5-s + (0.717 − 0.696i)6-s + 0.377·7-s + (−0.217 + 0.976i)8-s + (0.116 + 0.993i)9-s + (0.0892 − 0.0770i)10-s + (1.29 − 1.29i)11-s + (−0.642 − 0.766i)12-s + (0.759 + 0.759i)13-s + (0.0275 − 0.376i)14-s + (0.00691 + 0.117i)15-s + (0.957 + 0.288i)16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.810 + 0.586i$
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: \((2,\ 336,\ (\ :1/2),\ 0.810 + 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64188 - 0.531736i\)
\(L(\frac12)\) \(\approx\) \(1.64188 - 0.531736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.103 + 1.41i)T \)
3 \( 1 + (-1.29 - 1.15i)T \)
7 \( 1 - T \)
good5 \( 1 + (-0.186 - 0.186i)T + 5iT^{2} \)
11 \( 1 + (-4.29 + 4.29i)T - 11iT^{2} \)
13 \( 1 + (-2.73 - 2.73i)T + 13iT^{2} \)
17 \( 1 + 4.98iT - 17T^{2} \)
19 \( 1 + (3.09 - 3.09i)T - 19iT^{2} \)
23 \( 1 - 3.55iT - 23T^{2} \)
29 \( 1 + (-3.75 + 3.75i)T - 29iT^{2} \)
31 \( 1 - 6.58iT - 31T^{2} \)
37 \( 1 + (4.82 - 4.82i)T - 37iT^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 + (5.79 + 5.79i)T + 43iT^{2} \)
47 \( 1 - 3.22T + 47T^{2} \)
53 \( 1 + (7.95 + 7.95i)T + 53iT^{2} \)
59 \( 1 + (3.18 - 3.18i)T - 59iT^{2} \)
61 \( 1 + (3.17 + 3.17i)T + 61iT^{2} \)
67 \( 1 + (-2.39 + 2.39i)T - 67iT^{2} \)
71 \( 1 - 9.06iT - 71T^{2} \)
73 \( 1 - 2.32iT - 73T^{2} \)
79 \( 1 - 6.89iT - 79T^{2} \)
83 \( 1 + (1.12 + 1.12i)T + 83iT^{2} \)
89 \( 1 - 7.81T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−11.46217450825242355175848554457, −10.54670460446893313961679771413, −9.677115303037132273840535151983, −8.700500799403295386664696840538, −8.368044195323701684925265226482, −6.51994340841124663146057668228, −5.10060242848866081372216975044, −3.98907957434536545688207220965, −3.19996173587952695080612078421, −1.64406122717306217379341184467, 1.56973864700296261387978634870, 3.57977057914585049628053575997, 4.62508926731623769059689839288, 6.16648018130995322295757936412, 6.84038863681352663407379511989, 7.83833841241859704963992940868, 8.672875810484226280222113634295, 9.320775929348096798812496260734, 10.51865024314347814439564154460, 12.05725308286520949064580245475

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