Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.915 + 0.403i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.56 − 0.686i)2-s + (1.67 + 0.448i)3-s + (2.62 − 1.51i)4-s + (1.32 − 4.81i)5-s + 4.59·6-s + (0.158 + 6.99i)7-s + (−1.82 + 1.82i)8-s + (2.59 + 1.50i)9-s + (0.0981 − 13.2i)10-s + (−8.05 − 13.9i)11-s + (5.06 − 1.35i)12-s + (−0.148 + 0.148i)13-s + (5.20 + 17.8i)14-s + (4.38 − 7.46i)15-s + (−9.47 + 16.4i)16-s + (−5.20 + 19.4i)17-s + ⋯
L(s)  = 1  + (1.28 − 0.343i)2-s + (0.557 + 0.149i)3-s + (0.655 − 0.378i)4-s + (0.265 − 0.963i)5-s + 0.765·6-s + (0.0227 + 0.999i)7-s + (−0.227 + 0.227i)8-s + (0.288 + 0.166i)9-s + (0.00981 − 1.32i)10-s + (−0.732 − 1.26i)11-s + (0.422 − 0.113i)12-s + (−0.0114 + 0.0114i)13-s + (0.372 + 1.27i)14-s + (0.292 − 0.497i)15-s + (−0.592 + 1.02i)16-s + (−0.306 + 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.915 + 0.403i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.915 + 0.403i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.68871 - 0.565981i\)
\(L(\frac12)\)  \(\approx\)  \(2.68871 - 0.565981i\)
\(L(2)\)   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 \( 1 + (-1.67 - 0.448i)T \)
5 \( 1 + (-1.32 + 4.81i)T \)
7 \( 1 + (-0.158 - 6.99i)T \)
good2 \( 1 + (-2.56 + 0.686i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (8.05 + 13.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (0.148 - 0.148i)T - 169iT^{2} \)
17 \( 1 + (5.20 - 19.4i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-10.9 - 6.34i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (4.18 + 15.6i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 21.7iT - 841T^{2} \)
31 \( 1 + (-7.17 - 12.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (62.8 - 16.8i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 2.26T + 1.68e3T^{2} \)
43 \( 1 + (-35.1 + 35.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-55.8 + 14.9i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-101. - 27.1i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-59.5 + 34.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-17.3 + 30.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-8.56 + 31.9i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 59.5T + 5.04e3T^{2} \)
73 \( 1 + (-33.4 - 8.96i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-2.87 - 1.65i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-4.48 + 4.48i)T - 6.88e3iT^{2} \)
89 \( 1 + (123. + 71.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-113. - 113. i)T + 9.40e3iT^{2} \)
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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

−13.45175046275193394086152753333, −12.59823438331129271500945970959, −11.83662417656022782043621751604, −10.44815974093213007506119736261, −8.821111975027038601080001874692, −8.362362020786546799916029811098, −5.96162281204676136218783750954, −5.20610258265742845698603192364, −3.78559758248029575086391671830, −2.35428928339106949965242621736, 2.68498749607496045428889689772, 4.01252399084993605467640954993, 5.32124749353793488031267959353, 7.05266792571525234817555551908, 7.30298304641321490423992631060, 9.471441447554172646773501146701, 10.40946387067803993438819908083, 11.79450555623191581811932404229, 13.04504305185924413570681886339, 13.73709862355139200147744462079

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