Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Sign $0.376 + 0.926i$
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 + (−1.50 − 1.65i)5-s + (2.20 + 1.46i)7-s − 1.00i·9-s + 1.46·11-s + (0.887 − 0.887i)13-s + (−2.23 − 0.103i)15-s + (−2.10 − 2.10i)17-s + 3.95·19-s + (2.59 − 0.526i)21-s + (4.13 + 4.13i)23-s + (−0.462 + 4.97i)25-s + (−0.707 − 0.707i)27-s − 5.18i·29-s − 6.10i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.673 − 0.739i)5-s + (0.833 + 0.552i)7-s − 0.333i·9-s + 0.441·11-s + (0.246 − 0.246i)13-s + (−0.576 − 0.0267i)15-s + (−0.510 − 0.510i)17-s + 0.908·19-s + (0.565 − 0.114i)21-s + (0.861 + 0.861i)23-s + (−0.0925 + 0.995i)25-s + (−0.136 − 0.136i)27-s − 0.962i·29-s − 1.09i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.376 + 0.926i$
motivic weight  =  \(1\)
character  :  $\chi_{1680} (433, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1680,\ (\ :1/2),\ 0.376 + 0.926i)\)
\(L(1)\)  \(\approx\)  \(1.975030113\)
\(L(\frac12)\)  \(\approx\)  \(1.975030113\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.50 + 1.65i)T \)
7 \( 1 + (-2.20 - 1.46i)T \)
good11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + (-0.887 + 0.887i)T - 13iT^{2} \)
17 \( 1 + (2.10 + 2.10i)T + 17iT^{2} \)
19 \( 1 - 3.95T + 19T^{2} \)
23 \( 1 + (-4.13 - 4.13i)T + 23iT^{2} \)
29 \( 1 + 5.18iT - 29T^{2} \)
31 \( 1 + 6.10iT - 31T^{2} \)
37 \( 1 + (-2.25 + 2.25i)T - 37iT^{2} \)
41 \( 1 + 0.769iT - 41T^{2} \)
43 \( 1 + (-5.18 - 5.18i)T + 43iT^{2} \)
47 \( 1 + (8.57 + 8.57i)T + 47iT^{2} \)
53 \( 1 + (0.544 + 0.544i)T + 53iT^{2} \)
59 \( 1 - 3.19T + 59T^{2} \)
61 \( 1 + 1.42iT - 61T^{2} \)
67 \( 1 + (-5.93 + 5.93i)T - 67iT^{2} \)
71 \( 1 + 7.62T + 71T^{2} \)
73 \( 1 + (-6.81 + 6.81i)T - 73iT^{2} \)
79 \( 1 - 4.52iT - 79T^{2} \)
83 \( 1 + (-6.75 + 6.75i)T - 83iT^{2} \)
89 \( 1 - 1.19T + 89T^{2} \)
97 \( 1 + (8.68 + 8.68i)T + 97iT^{2} \)
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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

−9.200868645370788728467510859260, −8.274497096305548291138502841039, −7.81319296822253865529647039872, −7.03711109731225314218181592604, −5.85860587298693717787448299344, −5.05975148651272634143789810103, −4.20267284556659502241031055175, −3.18432987282353139826934682865, −1.96199282817752544882982398097, −0.834172367789482434231269979883, 1.29101926451057473091092356813, 2.70802627485331746646582743236, 3.63914112720517828083773685056, 4.37169567019189954286039121515, 5.19252909242407984762190193252, 6.57114764670937299145865532503, 7.11373760230580081571518093482, 8.004083576310768605506292214559, 8.608058198206992826255080156096, 9.440221256718840550841204307383

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