Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.298 - 0.954i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.60 + 2.77i)2-s + (−0.199 − 2.99i)3-s + (−3.15 + 5.45i)4-s + (4.78 + 1.45i)5-s + (7.99 − 5.35i)6-s + (6.02 + 3.55i)7-s − 7.38·8-s + (−8.92 + 1.19i)9-s + (3.61 + 15.6i)10-s + (−10.5 − 6.09i)11-s + (16.9 + 8.34i)12-s + 8.47i·13-s + (−0.220 + 22.4i)14-s + (3.41 − 14.6i)15-s + (0.744 + 1.29i)16-s + (5.29 − 9.17i)17-s + ⋯
L(s)  = 1  + (0.802 + 1.38i)2-s + (−0.0666 − 0.997i)3-s + (−0.787 + 1.36i)4-s + (0.956 + 0.291i)5-s + (1.33 − 0.893i)6-s + (0.861 + 0.508i)7-s − 0.923·8-s + (−0.991 + 0.132i)9-s + (0.361 + 1.56i)10-s + (−0.959 − 0.553i)11-s + (1.41 + 0.695i)12-s + 0.651i·13-s + (−0.0157 + 1.60i)14-s + (0.227 − 0.973i)15-s + (0.0465 + 0.0806i)16-s + (0.311 − 0.539i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.298 - 0.954i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (74, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.298 - 0.954i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.73170 + 1.27335i\)
\(L(\frac12)\)  \(\approx\)  \(1.73170 + 1.27335i\)
\(L(2)\)   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 \( 1 + (0.199 + 2.99i)T \)
5 \( 1 + (-4.78 - 1.45i)T \)
7 \( 1 + (-6.02 - 3.55i)T \)
good2 \( 1 + (-1.60 - 2.77i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (10.5 + 6.09i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 8.47iT - 169T^{2} \)
17 \( 1 + (-5.29 + 9.17i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (10.0 + 17.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (15.2 + 26.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 42.8iT - 841T^{2} \)
31 \( 1 + (6.11 - 10.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (28.8 - 16.6i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 6.40iT - 1.68e3T^{2} \)
43 \( 1 + 20.0iT - 1.84e3T^{2} \)
47 \( 1 + (11.8 + 20.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (43.9 - 76.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-41.9 - 24.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-10.4 - 18.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (19.6 + 11.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 2.44iT - 5.04e3T^{2} \)
73 \( 1 + (-76.5 - 44.2i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (3.20 + 5.54i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 103.T + 6.88e3T^{2} \)
89 \( 1 + (-54.2 + 31.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 140. iT - 9.40e3T^{2} \)
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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

−13.84868364361459217410742689722, −13.22134199051725437831689221353, −12.05610711048707337549624346856, −10.74682238585303641001095196301, −8.806224355001647837179312634140, −7.892142547571631485596850181411, −6.74857589042816690869247084339, −5.86727322048537415624661633355, −4.93822271776651067376888009269, −2.38768769752605970640162221681, 1.87537482409708355886494734665, 3.54255141533522830104934663999, 4.87433479071234937493047838103, 5.56799047494503068834619848865, 8.090455855216911579412532849859, 9.682764163230745497389649565429, 10.38899525944730261452917194790, 10.97688472744519739697914493651, 12.30118377059369344272669435504, 13.17686888371783482904637700389

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