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Properties

Label 2-1680-7.2-c1-0-31
Degree $2$
Conductor $1680$
Sign $-0.749 + 0.661i$
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  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−1.62 − 2.09i)7-s + (−0.499 + 0.866i)9-s + (−2.12 − 3.67i)11-s + 3.24·13-s + 0.999·15-s + (−2.12 − 3.67i)17-s + (−3.5 + 6.06i)19-s + (0.999 − 2.44i)21-s + (−2.12 + 3.67i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s − 1.75·29-s + (−4.74 − 8.21i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.612 − 0.790i)7-s + (−0.166 + 0.288i)9-s + (−0.639 − 1.10i)11-s + 0.899·13-s + 0.258·15-s + (−0.514 − 0.891i)17-s + (−0.802 + 1.39i)19-s + (0.218 − 0.534i)21-s + (−0.442 + 0.766i)23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s − 0.326·29-s + (−0.851 − 1.47i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.749 + 0.661i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7094725656\)
\(L(\frac12)\) \(\approx\) \(0.7094725656\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.62 + 2.09i)T \)
good11 \( 1 + (2.12 + 3.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.24T + 13T^{2} \)
17 \( 1 + (2.12 + 3.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.12 - 3.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.75T + 29T^{2} \)
31 \( 1 + (4.74 + 8.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.62 - 2.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 3.24T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.24 + 7.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.12 + 8.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.24 - 3.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.62 + 4.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (4.62 + 8.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (-5.12 + 8.87i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.485T + 97T^{2} \)
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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

−9.182708807168847205953428027059, −8.202474875641004376016605506995, −7.74703148828337891491615907407, −6.43888056740140074101200871595, −5.86225775703433962826963990304, −4.86681044233011888321634773242, −3.78438793808603064515162068640, −3.29608244239695039311081512754, −1.83808710557813370129038928978, −0.23761505732460834420497445216, 1.84197818406225752534186192206, 2.54942453819010427698374911348, 3.58425650033496315336690991275, 4.74854889558621941820271191802, 5.76975472520627576144018221744, 6.63184565245481584152736115493, 7.02116535960512809784683505371, 8.242780864949757746398277120197, 8.788160573508715414694400675013, 9.517526980740991201464557942073

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