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Properties

Label 2-1680-35.27-c1-0-41
Degree $2$
Conductor $1680$
Sign $0.476 + 0.879i$
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  + (0.707 + 0.707i)3-s + (−1.35 + 1.77i)5-s + (0.498 − 2.59i)7-s + 1.00i·9-s + 1.97·11-s + (−4.59 − 4.59i)13-s + (−2.21 + 0.295i)15-s + (4.07 − 4.07i)17-s + 0.683·19-s + (2.18 − 1.48i)21-s + (−5.64 + 5.64i)23-s + (−1.31 − 4.82i)25-s + (−0.707 + 0.707i)27-s − 5.55i·29-s − 6.08i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.607 + 0.794i)5-s + (0.188 − 0.982i)7-s + 0.333i·9-s + 0.594·11-s + (−1.27 − 1.27i)13-s + (−0.572 + 0.0763i)15-s + (0.988 − 0.988i)17-s + 0.156·19-s + (0.477 − 0.324i)21-s + (−1.17 + 1.17i)23-s + (−0.262 − 0.964i)25-s + (−0.136 + 0.136i)27-s − 1.03i·29-s − 1.09i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.476 + 0.879i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.476 + 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.379782470\)
\(L(\frac12)\) \(\approx\) \(1.379782470\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (1.35 - 1.77i)T \)
7 \( 1 + (-0.498 + 2.59i)T \)
good11 \( 1 - 1.97T + 11T^{2} \)
13 \( 1 + (4.59 + 4.59i)T + 13iT^{2} \)
17 \( 1 + (-4.07 + 4.07i)T - 17iT^{2} \)
19 \( 1 - 0.683T + 19T^{2} \)
23 \( 1 + (5.64 - 5.64i)T - 23iT^{2} \)
29 \( 1 + 5.55iT - 29T^{2} \)
31 \( 1 + 6.08iT - 31T^{2} \)
37 \( 1 + (-2.16 - 2.16i)T + 37iT^{2} \)
41 \( 1 + 9.16iT - 41T^{2} \)
43 \( 1 + (0.140 - 0.140i)T - 43iT^{2} \)
47 \( 1 + (1.75 - 1.75i)T - 47iT^{2} \)
53 \( 1 + (0.681 - 0.681i)T - 53iT^{2} \)
59 \( 1 - 9.12T + 59T^{2} \)
61 \( 1 + 2.91iT - 61T^{2} \)
67 \( 1 + (-7.72 - 7.72i)T + 67iT^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + (7.15 + 7.15i)T + 73iT^{2} \)
79 \( 1 + 12.3iT - 79T^{2} \)
83 \( 1 + (7.79 + 7.79i)T + 83iT^{2} \)
89 \( 1 - 9.74T + 89T^{2} \)
97 \( 1 + (-0.937 + 0.937i)T - 97iT^{2} \)
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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

−9.553800835679237047378832412491, −8.093211292233024963051942356497, −7.63342393400988525831554325700, −7.19581959780022886926393250609, −5.95462598972785880244276873909, −4.97536961463530636100893918490, −3.96514997454727871921479160979, −3.37194523157598936413841766363, −2.34520125172743810393532407140, −0.51417950917264434498513253433, 1.38994238480787659662247165406, 2.32942842499363437928396406921, 3.59686178545555717103587667978, 4.50943293793105105856915827710, 5.31963012880421555976220501997, 6.37412770281767744784579566278, 7.14871394504156965367269936059, 8.192358028918983099422157906459, 8.500225673804346069687516380248, 9.374095154035512096798881198854

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