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Properties

Label 2-105-21.5-c3-0-3
Degree $2$
Conductor $105$
Sign $-0.836 - 0.547i$
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  + (−0.133 − 0.0768i)2-s + (2.47 + 4.57i)3-s + (−3.98 − 6.90i)4-s + (−2.5 + 4.33i)5-s + (0.0223 − 0.798i)6-s + (−15.0 + 10.7i)7-s + 2.45i·8-s + (−14.7 + 22.5i)9-s + (0.665 − 0.384i)10-s + (−14.9 + 8.65i)11-s + (21.7 − 35.2i)12-s + 36.4i·13-s + (2.83 − 0.281i)14-s + (−25.9 − 0.727i)15-s + (−31.7 + 54.9i)16-s + (−14.8 − 25.7i)17-s + ⋯
L(s)  = 1  + (−0.0470 − 0.0271i)2-s + (0.475 + 0.879i)3-s + (−0.498 − 0.863i)4-s + (−0.223 + 0.387i)5-s + (0.00152 − 0.0543i)6-s + (−0.812 + 0.583i)7-s + 0.108i·8-s + (−0.547 + 0.836i)9-s + (0.0210 − 0.0121i)10-s + (−0.410 + 0.237i)11-s + (0.522 − 0.849i)12-s + 0.778i·13-s + (0.0540 − 0.00537i)14-s + (−0.447 − 0.0125i)15-s + (−0.495 + 0.858i)16-s + (−0.212 − 0.367i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.836 - 0.547i$
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: \((2,\ 105,\ (\ :3/2),\ -0.836 - 0.547i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.218172 + 0.731913i\)
\(L(\frac12)\) \(\approx\) \(0.218172 + 0.731913i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.47 - 4.57i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (15.0 - 10.7i)T \)
good2 \( 1 + (0.133 + 0.0768i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (14.9 - 8.65i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 36.4iT - 2.19e3T^{2} \)
17 \( 1 + (14.8 + 25.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (112. + 65.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-134. - 77.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 165. iT - 2.43e4T^{2} \)
31 \( 1 + (-12.9 + 7.50i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (16.8 - 29.2i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 274.T + 6.89e4T^{2} \)
43 \( 1 - 248.T + 7.95e4T^{2} \)
47 \( 1 + (-229. + 397. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (211. - 122. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-292. - 506. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (221. + 128. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (137. + 237. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.06e3iT - 3.57e5T^{2} \)
73 \( 1 + (861. - 497. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (5.81 - 10.0i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 584.T + 5.71e5T^{2} \)
89 \( 1 + (548. - 950. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.11e3iT - 9.12e5T^{2} \)
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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

−13.86620806520873817820670777375, −12.90077434586942359981949778609, −11.23345747425022177367233768522, −10.41902853825199784001994148171, −9.320368673418480760849266330091, −8.790792343383772680708055164237, −6.89461112637243590647470588800, −5.45980104584481433505903619351, −4.24476112687880587828315810851, −2.59099244195467658376495510882, 0.39762825238240816631985557114, 2.87687890050076087305156414981, 4.15994730417701794849328524886, 6.19986252976120787020334877897, 7.51713918752704927442877370000, 8.290443502022280287563757483823, 9.270637831515586704750359367749, 10.75818052310214527093944380043, 12.38544977457609771946024014900, 12.85107467369870109803072847250

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