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Properties

Label 2-105-21.5-c3-0-1
Degree $2$
Conductor $105$
Sign $-0.141 - 0.989i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.48 − 0.859i)2-s + (−4.80 + 1.98i)3-s + (−2.52 − 4.36i)4-s + (2.5 − 4.33i)5-s + (8.85 + 1.17i)6-s + (2.28 − 18.3i)7-s + 22.4i·8-s + (19.1 − 19.0i)9-s + (−7.44 + 4.29i)10-s + (−49.7 + 28.7i)11-s + (20.7 + 15.9i)12-s + 44.6i·13-s + (−19.2 + 25.4i)14-s + (−3.41 + 25.7i)15-s + (−0.881 + 1.52i)16-s + (54.8 + 95.0i)17-s + ⋯
L(s)  = 1  + (−0.526 − 0.304i)2-s + (−0.924 + 0.381i)3-s + (−0.315 − 0.545i)4-s + (0.223 − 0.387i)5-s + (0.602 + 0.0798i)6-s + (0.123 − 0.992i)7-s + 0.991i·8-s + (0.708 − 0.705i)9-s + (−0.235 + 0.135i)10-s + (−1.36 + 0.787i)11-s + (0.499 + 0.384i)12-s + 0.952i·13-s + (−0.366 + 0.485i)14-s + (−0.0587 + 0.443i)15-s + (−0.0137 + 0.0238i)16-s + (0.783 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.141 - 0.989i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.141 - 0.989i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.177709 + 0.204839i\)
\(L(\frac12)\) \(\approx\) \(0.177709 + 0.204839i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.80 - 1.98i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 + (-2.28 + 18.3i)T \)
good2 \( 1 + (1.48 + 0.859i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (49.7 - 28.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 44.6iT - 2.19e3T^{2} \)
17 \( 1 + (-54.8 - 95.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (79.5 + 45.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-27.1 - 15.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 183. iT - 2.43e4T^{2} \)
31 \( 1 + (-123. + 71.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (114. - 199. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 185.T + 6.89e4T^{2} \)
43 \( 1 + 22.3T + 7.95e4T^{2} \)
47 \( 1 + (-116. + 201. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (400. - 230. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (229. + 397. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (248. + 143. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-416. - 720. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 471. iT - 3.57e5T^{2} \)
73 \( 1 + (-469. + 270. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (175. - 304. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 866.T + 5.71e5T^{2} \)
89 \( 1 + (519. - 899. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 323. iT - 9.12e5T^{2} \)
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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

−13.41003656714221298952845245437, −12.44261425362308918495513353255, −11.02116904141101147424477828058, −10.38924085927490675986865175372, −9.682982738155457457386597333941, −8.275190720629787277614720746996, −6.72009034499122557823210440416, −5.27967064457765952620735302371, −4.38170278844760314725167820631, −1.49042969576728664287727997863, 0.20958881213672679229922874231, 2.84705672969321052425032728535, 5.11000249089642968007169245434, 6.12447077548112687156735147201, 7.57698585791643134455352192882, 8.332083646582808475539558498890, 9.826694634114265475706547803545, 10.82111481165545497694319951669, 12.06631558256229816663598616276, 12.82967657503660617279396644294

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