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Properties

Label 2-1680-35.13-c1-0-21
Degree $2$
Conductor $1680$
Sign $0.857 - 0.514i$
Analytic cond. $13.4148$
Root an. cond. $3.66263$
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  + (−0.707 + 0.707i)3-s + (−1.43 + 1.71i)5-s + (2.23 + 1.41i)7-s − 1.00i·9-s − 0.566·11-s + (5.03 − 5.03i)13-s + (−0.194 − 2.22i)15-s + (−0.984 − 0.984i)17-s + 7.61·19-s + (−2.58 + 0.581i)21-s + (−2.55 − 2.55i)23-s + (−0.865 − 4.92i)25-s + (0.707 + 0.707i)27-s − 7.85i·29-s + 7.09i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.642 + 0.765i)5-s + (0.845 + 0.534i)7-s − 0.333i·9-s − 0.170·11-s + (1.39 − 1.39i)13-s + (−0.0501 − 0.575i)15-s + (−0.238 − 0.238i)17-s + 1.74·19-s + (−0.563 + 0.126i)21-s + (−0.531 − 0.531i)23-s + (−0.173 − 0.984i)25-s + (0.136 + 0.136i)27-s − 1.45i·29-s + 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.857 - 0.514i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.857 - 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.561637336\)
\(L(\frac12)\) \(\approx\) \(1.561637336\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.43 - 1.71i)T \)
7 \( 1 + (-2.23 - 1.41i)T \)
good11 \( 1 + 0.566T + 11T^{2} \)
13 \( 1 + (-5.03 + 5.03i)T - 13iT^{2} \)
17 \( 1 + (0.984 + 0.984i)T + 17iT^{2} \)
19 \( 1 - 7.61T + 19T^{2} \)
23 \( 1 + (2.55 + 2.55i)T + 23iT^{2} \)
29 \( 1 + 7.85iT - 29T^{2} \)
31 \( 1 - 7.09iT - 31T^{2} \)
37 \( 1 + (-0.887 + 0.887i)T - 37iT^{2} \)
41 \( 1 + 6.29iT - 41T^{2} \)
43 \( 1 + (-2.74 - 2.74i)T + 43iT^{2} \)
47 \( 1 + (-3.25 - 3.25i)T + 47iT^{2} \)
53 \( 1 + (-7.27 - 7.27i)T + 53iT^{2} \)
59 \( 1 - 6.62T + 59T^{2} \)
61 \( 1 + 5.46iT - 61T^{2} \)
67 \( 1 + (-2.53 + 2.53i)T - 67iT^{2} \)
71 \( 1 + 8.01T + 71T^{2} \)
73 \( 1 + (8.97 - 8.97i)T - 73iT^{2} \)
79 \( 1 - 1.85iT - 79T^{2} \)
83 \( 1 + (2.61 - 2.61i)T - 83iT^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + (-4.32 - 4.32i)T + 97iT^{2} \)
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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

−9.462308637156942278793367509629, −8.457259525564288202917765644978, −7.927594236383676111330243712186, −7.11708964261657346673725213416, −5.94874547450182259435190843803, −5.49588861902981039973001105773, −4.40083600693371842622711134510, −3.48116317121042809247880849937, −2.61370310430780930075062882099, −0.899941465588113172856184528506, 0.989052049621424967270695646541, 1.75356562606334450484790048270, 3.57797540418667636248775386276, 4.29344444087668630413004433091, 5.16240853246793418462946420486, 5.97264024046127814871399113524, 7.11278908095069902022581265297, 7.60451582899748041666606605811, 8.495934726386634785529547332654, 9.049275680081704188355804466043

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