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Properties

Label 2-1680-7.2-c1-0-6
Degree $2$
Conductor $1680$
Sign $-0.895 - 0.444i$
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 + (−1.32 + 2.29i)7-s + (−0.499 + 0.866i)9-s + (1.82 + 3.15i)11-s + 2.64·13-s − 0.999·15-s + (−1.82 − 3.15i)17-s + (−1.14 + 1.98i)19-s − 2.64·21-s + (−1.82 + 3.15i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s + 2.35·29-s + (3.14 + 5.44i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.223 + 0.387i)5-s + (−0.499 + 0.866i)7-s + (−0.166 + 0.288i)9-s + (0.549 + 0.951i)11-s + 0.733·13-s − 0.258·15-s + (−0.442 − 0.765i)17-s + (−0.262 + 0.455i)19-s − 0.577·21-s + (−0.380 + 0.658i)23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s + 0.437·29-s + (0.564 + 0.978i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.895 - 0.444i$
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: \((2,\ 1680,\ (\ :1/2),\ -0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.271134513\)
\(L(\frac12)\) \(\approx\) \(1.271134513\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 2.29i)T \)
good11 \( 1 + (-1.82 - 3.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 + (1.82 + 3.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.14 - 1.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.82 - 3.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.35T + 29T^{2} \)
31 \( 1 + (-3.14 - 5.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.32 + 4.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 9.93T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.64 - 6.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.46 + 4.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.32 + 4.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.35T + 71T^{2} \)
73 \( 1 + (-6.61 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.14 - 7.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.93T + 83T^{2} \)
89 \( 1 + (6.11 - 10.5i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.770040184864350547024168807273, −8.852417781588933446266027467745, −8.375648788705750999532257568038, −7.19249357678600856467297150821, −6.55757896427041654875285879749, −5.61902850198731708751323307454, −4.66553265652863173948420701097, −3.72335568923324631890558115967, −2.89933183256941281117138411791, −1.79512940376287117020329961585, 0.46412967758151667315342979580, 1.59504549541270286438718596024, 3.07995307619936354304562217384, 3.85251830300451391227541910935, 4.69785779078396905786769235119, 6.23016825959259340465513668860, 6.38552351217793232068950119647, 7.46603162738979841506869650593, 8.457075216918978710415402422719, 8.651981482294747018407964358195

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