Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.243 + 0.970i$
Motivic weight 1
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 + (−2.23 − 0.0836i)5-s + (0.0627 + 2.64i)7-s + 1.00i·9-s − 3.98·11-s + (−0.500 − 0.500i)13-s + (−1.52 − 1.63i)15-s + (1.67 − 1.67i)17-s − 7.21·19-s + (−1.82 + 1.91i)21-s + (5.16 − 5.16i)23-s + (4.98 + 0.373i)25-s + (−0.707 + 0.707i)27-s − 3.65i·29-s − 4.93i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.999 − 0.0373i)5-s + (0.0237 + 0.999i)7-s + 0.333i·9-s − 1.20·11-s + (−0.138 − 0.138i)13-s + (−0.392 − 0.423i)15-s + (0.407 − 0.407i)17-s − 1.65·19-s + (−0.398 + 0.417i)21-s + (1.07 − 1.07i)23-s + (0.997 + 0.0747i)25-s + (−0.136 + 0.136i)27-s − 0.678i·29-s − 0.886i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.243 + 0.970i$
motivic weight  =  \(1\)
character  :  $\chi_{1680} (97, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1680,\ (\ :1/2),\ -0.243 + 0.970i)\)
\(L(1)\)  \(\approx\)  \(0.4542524740\)
\(L(\frac12)\)  \(\approx\)  \(0.4542524740\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (2.23 + 0.0836i)T \)
7 \( 1 + (-0.0627 - 2.64i)T \)
good11 \( 1 + 3.98T + 11T^{2} \)
13 \( 1 + (0.500 + 0.500i)T + 13iT^{2} \)
17 \( 1 + (-1.67 + 1.67i)T - 17iT^{2} \)
19 \( 1 + 7.21T + 19T^{2} \)
23 \( 1 + (-5.16 + 5.16i)T - 23iT^{2} \)
29 \( 1 + 3.65iT - 29T^{2} \)
31 \( 1 + 4.93iT - 31T^{2} \)
37 \( 1 + (-0.292 - 0.292i)T + 37iT^{2} \)
41 \( 1 + 7.63iT - 41T^{2} \)
43 \( 1 + (3.65 - 3.65i)T - 43iT^{2} \)
47 \( 1 + (0.305 - 0.305i)T - 47iT^{2} \)
53 \( 1 + (-5.39 + 5.39i)T - 53iT^{2} \)
59 \( 1 - 6.10T + 59T^{2} \)
61 \( 1 - 7.11iT - 61T^{2} \)
67 \( 1 + (0.944 + 0.944i)T + 67iT^{2} \)
71 \( 1 + 1.19T + 71T^{2} \)
73 \( 1 + (1.38 + 1.38i)T + 73iT^{2} \)
79 \( 1 + 8.64iT - 79T^{2} \)
83 \( 1 + (11.9 + 11.9i)T + 83iT^{2} \)
89 \( 1 + 7.82T + 89T^{2} \)
97 \( 1 + (7.43 - 7.43i)T - 97iT^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.846260630197062602823625486653, −8.444678678547014886683421988149, −7.75894742773483494250062629977, −6.84382069170238872643508757898, −5.72085369583022399136679814578, −4.88352958711504245274476854582, −4.14356760397704045977489908542, −2.95266493243527068522046043154, −2.33823257479644297215881010444, −0.16759613966904498662745504018, 1.30892733756317159385650168993, 2.76973325902893529008630620321, 3.62903470780896998171910279158, 4.47547989334331794444073723995, 5.40767203971729299918984049729, 6.81335674615289254722060333204, 7.15041933669602068515894740316, 8.124548667924828113608949816796, 8.397014410712015120627376497063, 9.556296832680223948328622946161

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