Properties

Label 2-105-105.44-c2-0-19
Degree $2$
Conductor $105$
Sign $0.298 + 0.954i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.298 + 0.954i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.298 + 0.954i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.73170 - 1.27335i\)
\(L(\frac12)\) \(\approx\) \(1.73170 - 1.27335i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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