Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.616 - 0.787i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.644 − 0.372i)2-s + (−2.67 − 1.35i)3-s + (−1.72 − 2.98i)4-s + (1.93 + 1.11i)5-s + (1.22 + 1.87i)6-s + (−5.98 + 3.63i)7-s + 5.54i·8-s + (5.32 + 7.25i)9-s + (−0.832 − 1.44i)10-s + (−8.35 + 4.82i)11-s + (0.565 + 10.3i)12-s − 0.0661·13-s + (5.21 − 0.116i)14-s + (−3.66 − 5.61i)15-s + (−4.82 + 8.36i)16-s + (−28.1 + 16.2i)17-s + ⋯
L(s)  = 1  + (−0.322 − 0.186i)2-s + (−0.892 − 0.451i)3-s + (−0.430 − 0.746i)4-s + (0.387 + 0.223i)5-s + (0.203 + 0.311i)6-s + (−0.854 + 0.519i)7-s + 0.692i·8-s + (0.591 + 0.806i)9-s + (−0.0832 − 0.144i)10-s + (−0.759 + 0.438i)11-s + (0.0471 + 0.860i)12-s − 0.00508·13-s + (0.372 − 0.00832i)14-s + (−0.244 − 0.374i)15-s + (−0.301 + 0.522i)16-s + (−1.65 + 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.616 - 0.787i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (86, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.616 - 0.787i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0383419 + 0.0786984i\)
\(L(\frac12)\)  \(\approx\)  \(0.0383419 + 0.0786984i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (2.67 + 1.35i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (5.98 - 3.63i)T \)
good2 \( 1 + (0.644 + 0.372i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (8.35 - 4.82i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 0.0661T + 169T^{2} \)
17 \( 1 + (28.1 - 16.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-12.7 + 22.1i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (8.49 + 4.90i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 6.58iT - 841T^{2} \)
31 \( 1 + (16.4 + 28.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (27.3 - 47.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 14.8iT - 1.68e3T^{2} \)
43 \( 1 + 14.3T + 1.84e3T^{2} \)
47 \( 1 + (63.7 + 36.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-44.3 + 25.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (75.2 - 43.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (12.5 - 21.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-24.0 - 41.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 113. iT - 5.04e3T^{2} \)
73 \( 1 + (21.2 + 36.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-9.49 + 16.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 46.8iT - 6.88e3T^{2} \)
89 \( 1 + (-45.4 - 26.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 147.T + 9.40e3T^{2} \)
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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

−13.42084182802673271725297728408, −13.08396975356698973642314253222, −11.64135237092737744250821501996, −10.62415539494770580455916069646, −9.846047679883862032874983537394, −8.667923789694420929122071358224, −6.89888619829327306990524214474, −5.94293784666458194333262650084, −4.84061845509551743889671326302, −2.14640022061172408000359338128, 0.07687963423398083583005812465, 3.50098952027044011033442106625, 4.89094847233926606048425057665, 6.32895778849930719869458711970, 7.49001521062990499197183245103, 9.027028412911677868069946238997, 9.842056838216550428351332979275, 10.90462919813595850296122125600, 12.20884542443227979141275379002, 13.06321940099011161373531236400

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