Properties

Label 2-1680-7.2-c1-0-16
Degree $2$
Conductor $1680$
Sign $0.358 - 0.933i$
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 + (2.40 + 1.10i)7-s + (−0.499 + 0.866i)9-s + (1.90 + 3.29i)11-s + 4.31·13-s − 0.999·15-s + (0.657 + 1.13i)17-s + (3.81 − 6.60i)19-s + (0.247 + 2.63i)21-s + (3.41 − 5.90i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s − 4.30·29-s + (0.747 + 1.29i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.908 + 0.416i)7-s + (−0.166 + 0.288i)9-s + (0.574 + 0.994i)11-s + 1.19·13-s − 0.258·15-s + (0.159 + 0.276i)17-s + (0.875 − 1.51i)19-s + (0.0539 + 0.574i)21-s + (0.711 − 1.23i)23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s − 0.799·29-s + (0.134 + 0.232i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.358 - 0.933i$
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: \((2,\ 1680,\ (\ :1/2),\ 0.358 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.254454416\)
\(L(\frac12)\) \(\approx\) \(2.254454416\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.40 - 1.10i)T \)
good11 \( 1 + (-1.90 - 3.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.31T + 13T^{2} \)
17 \( 1 + (-0.657 - 1.13i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.81 + 6.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.41 + 5.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.30T + 29T^{2} \)
31 \( 1 + (-0.747 - 1.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.65 - 8.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 4.80T + 43T^{2} \)
47 \( 1 + (1.75 - 3.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.06 - 7.04i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.15 - 7.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.31 + 5.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.72 + 13.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.32T + 71T^{2} \)
73 \( 1 + (1.91 + 3.30i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.06 - 1.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + (5.65 - 9.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−9.303696329113422071325918933683, −8.816859120529012965890640933015, −8.043933378631384280096000903242, −7.12278922293906688520179890514, −6.39498501118532769129292762765, −5.16826684359226789435974501316, −4.58877387988715818033048738077, −3.59416034837587283790702894997, −2.59781697443951232340216406709, −1.38793272217856040307081568683, 1.00136674719589625973213668074, 1.72418931001745979105539161450, 3.50691588749174051349789040845, 3.81113829906137293701695373592, 5.32352956897425091413426077296, 5.80400065145272947345363232696, 7.00160505107213256562289742459, 7.68803568141425218135878344120, 8.434796060521803988844245170755, 8.904977014587105658425844795070

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