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Properties

Label 2-1680-21.20-c1-0-47
Degree $2$
Conductor $1680$
Sign $0.181 + 0.983i$
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  + (1.30 − 1.14i)3-s − 5-s + (1.35 − 2.27i)7-s + (0.392 − 2.97i)9-s + 5.73i·11-s − 2.24i·13-s + (−1.30 + 1.14i)15-s + 6.50·17-s + 0.217i·19-s + (−0.833 − 4.50i)21-s − 5.57i·23-s + 25-s + (−2.88 − 4.32i)27-s + 1.11i·29-s − 5.49i·31-s + ⋯
L(s)  = 1  + (0.751 − 0.659i)3-s − 0.447·5-s + (0.511 − 0.859i)7-s + (0.130 − 0.991i)9-s + 1.72i·11-s − 0.622i·13-s + (−0.336 + 0.294i)15-s + 1.57·17-s + 0.0499i·19-s + (−0.181 − 0.983i)21-s − 1.16i·23-s + 0.200·25-s + (−0.555 − 0.831i)27-s + 0.206i·29-s − 0.987i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.181 + 0.983i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.181 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.219752096\)
\(L(\frac12)\) \(\approx\) \(2.219752096\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 1.14i)T \)
5 \( 1 + T \)
7 \( 1 + (-1.35 + 2.27i)T \)
good11 \( 1 - 5.73iT - 11T^{2} \)
13 \( 1 + 2.24iT - 13T^{2} \)
17 \( 1 - 6.50T + 17T^{2} \)
19 \( 1 - 0.217iT - 19T^{2} \)
23 \( 1 + 5.57iT - 23T^{2} \)
29 \( 1 - 1.11iT - 29T^{2} \)
31 \( 1 + 5.49iT - 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 + 1.11T + 41T^{2} \)
43 \( 1 + 5.89T + 43T^{2} \)
47 \( 1 - 7.70T + 47T^{2} \)
53 \( 1 + 8.37iT - 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 1.33iT - 61T^{2} \)
67 \( 1 + 14.5T + 67T^{2} \)
71 \( 1 - 7.41iT - 71T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 + 4.54T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 - 4.45T + 89T^{2} \)
97 \( 1 - 9.33iT - 97T^{2} \)
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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

−9.143672610620195233427983996801, −7.980341709314796652351303564592, −7.73615671825429080816442342590, −7.11220673277807075058417460138, −6.13924690341341856709800683311, −4.82919530696034883168329233044, −4.10972767197748176835871913054, −3.11406296764810650985787739516, −1.98623947872682529808474731747, −0.855449411469866301572795712220, 1.42259849081919309886516551855, 2.89360654972713919241150137066, 3.41225500684917759086706151914, 4.47521669971828839360379657626, 5.44075037014406008298247850971, 6.05504823586024770708066169772, 7.56206737130231400731362073806, 8.006566603569994884111543517126, 8.835340209859176508342764261663, 9.254374828224862791407236107812

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