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Properties

Label 2-1680-7.2-c1-0-0
Degree $2$
Conductor $1680$
Sign $-0.701 - 0.712i$
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 + (2 − 1.73i)7-s + (−0.499 + 0.866i)9-s + (−2.5 − 4.33i)11-s − 5·13-s + 0.999·15-s + (2 + 3.46i)17-s + (−3.5 + 6.06i)19-s + (−2.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + 0.999·27-s + (−1 − 1.73i)31-s + (−2.5 + 4.33i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.755 − 0.654i)7-s + (−0.166 + 0.288i)9-s + (−0.753 − 1.30i)11-s − 1.38·13-s + 0.258·15-s + (0.485 + 0.840i)17-s + (−0.802 + 1.39i)19-s + (−0.545 − 0.188i)21-s + (0.104 − 0.180i)23-s + (−0.0999 − 0.173i)25-s + 0.192·27-s + (−0.179 − 0.311i)31-s + (−0.435 + 0.753i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.701 - 0.712i$
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: \((2,\ 1680,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1597855119\)
\(L(\frac12)\) \(\approx\) \(0.1597855119\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (5.5 - 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−9.875008337117172300089976161051, −8.520793907511696753380657272931, −7.921382372792288363445706629482, −7.48793522891975515262992772740, −6.39754611974417693666583111571, −5.68974836382059332668367513598, −4.76188174295819758240208273876, −3.74279348049847136373640752177, −2.65733814442995724574577673850, −1.45164570094999445784551396447, 0.06082354070551992358364437882, 1.98869150420860494146811865789, 2.84697869618795500349647197762, 4.37820587345781306290234564019, 5.03576302052680691499671645911, 5.29027389718820248794731076056, 6.83208236212602830687955128796, 7.42940549279839280970164998894, 8.330917458150082397045020997593, 9.133420932104137890376915145369

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