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Properties

Label 2-336-16.5-c1-0-0
Degree $2$
Conductor $336$
Sign $0.464 - 0.885i$
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  + (−0.244 − 1.39i)2-s + (−0.707 + 0.707i)3-s + (−1.88 + 0.680i)4-s + (−3.00 − 3.00i)5-s + (1.15 + 0.812i)6-s + i·7-s + (1.40 + 2.45i)8-s − 1.00i·9-s + (−3.44 + 4.91i)10-s + (3.36 + 3.36i)11-s + (0.848 − 1.81i)12-s + (−3.25 + 3.25i)13-s + (1.39 − 0.244i)14-s + 4.24·15-s + (3.07 − 2.56i)16-s + 2.73·17-s + ⋯
L(s)  = 1  + (−0.172 − 0.984i)2-s + (−0.408 + 0.408i)3-s + (−0.940 + 0.340i)4-s + (−1.34 − 1.34i)5-s + (0.472 + 0.331i)6-s + 0.377i·7-s + (0.497 + 0.867i)8-s − 0.333i·9-s + (−1.09 + 1.55i)10-s + (1.01 + 1.01i)11-s + (0.244 − 0.522i)12-s + (−0.902 + 0.902i)13-s + (0.372 − 0.0653i)14-s + 1.09·15-s + (0.768 − 0.640i)16-s + 0.663·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.464 - 0.885i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.464 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.309023 + 0.186805i\)
\(L(\frac12)\) \(\approx\) \(0.309023 + 0.186805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.244 + 1.39i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 - iT \)
good5 \( 1 + (3.00 + 3.00i)T + 5iT^{2} \)
11 \( 1 + (-3.36 - 3.36i)T + 11iT^{2} \)
13 \( 1 + (3.25 - 3.25i)T - 13iT^{2} \)
17 \( 1 - 2.73T + 17T^{2} \)
19 \( 1 + (3.56 - 3.56i)T - 19iT^{2} \)
23 \( 1 + 2.49iT - 23T^{2} \)
29 \( 1 + (6.70 - 6.70i)T - 29iT^{2} \)
31 \( 1 + 2.34T + 31T^{2} \)
37 \( 1 + (-2.74 - 2.74i)T + 37iT^{2} \)
41 \( 1 + 4.84iT - 41T^{2} \)
43 \( 1 + (0.394 + 0.394i)T + 43iT^{2} \)
47 \( 1 - 0.0322T + 47T^{2} \)
53 \( 1 + (6.62 + 6.62i)T + 53iT^{2} \)
59 \( 1 + (1.94 + 1.94i)T + 59iT^{2} \)
61 \( 1 + (4.32 - 4.32i)T - 61iT^{2} \)
67 \( 1 + (-1.83 + 1.83i)T - 67iT^{2} \)
71 \( 1 - 12.8iT - 71T^{2} \)
73 \( 1 - 5.10iT - 73T^{2} \)
79 \( 1 - 0.455T + 79T^{2} \)
83 \( 1 + (-1.92 + 1.92i)T - 83iT^{2} \)
89 \( 1 - 7.82iT - 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−11.88056101151539988012126315322, −11.04834374677005532010872358809, −9.723649893152008886084625691838, −9.146070345596522794299310194051, −8.288897594812855375453946421772, −7.17528634399749595134270812160, −5.25921755004424438509576802272, −4.41440138379495690078752479242, −3.77707315803520570752454511012, −1.61361768227473784890985972680, 0.29770401145800972161529147810, 3.27238174574150912460564853012, 4.32890937169488507396323187088, 5.87522783549102805026443194331, 6.70677614793965368804408458136, 7.57819271006686826839403635678, 8.008772782448392831860587108492, 9.440220805316817775940044584064, 10.64578146198686611385469790053, 11.25774619649568424119120502090

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