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Properties

Label 2-1680-28.3-c1-0-11
Degree $2$
Conductor $1680$
Sign $0.581 - 0.813i$
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 + (0.866 − 0.5i)5-s + (2.52 − 0.783i)7-s + (−0.499 − 0.866i)9-s + (3.57 + 2.06i)11-s + 6.61i·13-s + 0.999i·15-s + (0.894 + 0.516i)17-s + (−2.61 − 4.52i)19-s + (−0.585 + 2.58i)21-s + (−2.85 + 1.64i)23-s + (0.499 − 0.866i)25-s + 0.999·27-s + 2.30·29-s + (1.56 − 2.71i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.387 − 0.223i)5-s + (0.955 − 0.295i)7-s + (−0.166 − 0.288i)9-s + (1.07 + 0.622i)11-s + 1.83i·13-s + 0.258i·15-s + (0.217 + 0.125i)17-s + (−0.598 − 1.03i)19-s + (−0.127 + 0.563i)21-s + (−0.595 + 0.343i)23-s + (0.0999 − 0.173i)25-s + 0.192·27-s + 0.428·29-s + (0.281 − 0.488i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.581 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.936293341\)
\(L(\frac12)\) \(\approx\) \(1.936293341\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-2.52 + 0.783i)T \)
good11 \( 1 + (-3.57 - 2.06i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.61iT - 13T^{2} \)
17 \( 1 + (-0.894 - 0.516i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.61 + 4.52i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.85 - 1.64i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.30T + 29T^{2} \)
31 \( 1 + (-1.56 + 2.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.14 - 7.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.90iT - 41T^{2} \)
43 \( 1 - 0.130iT - 43T^{2} \)
47 \( 1 + (5.85 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.829 + 1.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.28 - 3.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.22 + 4.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.66 - 3.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.00iT - 71T^{2} \)
73 \( 1 + (0.175 + 0.101i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.06 + 4.07i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + (13.0 - 7.54i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.29iT - 97T^{2} \)
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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

−9.527376392146728046775300439427, −8.819200007853593982874645070775, −8.024172168825843580025816598295, −6.76118026889966857959080694877, −6.48907543123567235235549185792, −5.14761417397206032851378108683, −4.47033070148446324129204258399, −3.94003744716149074761673065486, −2.24178092054382943937978535616, −1.31448576896490345338927803320, 0.878130072206641989475678996890, 1.98136455005909158861555495399, 3.11686765001514129091127065119, 4.23700226827372907111906947209, 5.46650740059775332873215552833, 5.86632477485670159195715628980, 6.72985895295557832655239096689, 7.88396358543588156266320797936, 8.197727240927520234732710743084, 9.128889352239900139547391682654

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