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Properties

Label 2-105-21.5-c3-0-6
Degree $2$
Conductor $105$
Sign $0.972 - 0.231i$
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  + (−1.35 − 0.781i)2-s + (−5.18 + 0.383i)3-s + (−2.77 − 4.81i)4-s + (−2.5 + 4.33i)5-s + (7.31 + 3.52i)6-s + (−17.7 + 5.23i)7-s + 21.1i·8-s + (26.7 − 3.97i)9-s + (6.76 − 3.90i)10-s + (26.1 − 15.0i)11-s + (16.2 + 23.8i)12-s − 59.7i·13-s + (28.1 + 6.80i)14-s + (11.2 − 23.3i)15-s + (−5.67 + 9.83i)16-s + (59.7 + 103. i)17-s + ⋯
L(s)  = 1  + (−0.478 − 0.276i)2-s + (−0.997 + 0.0738i)3-s + (−0.347 − 0.601i)4-s + (−0.223 + 0.387i)5-s + (0.497 + 0.240i)6-s + (−0.959 + 0.282i)7-s + 0.936i·8-s + (0.989 − 0.147i)9-s + (0.213 − 0.123i)10-s + (0.716 − 0.413i)11-s + (0.390 + 0.574i)12-s − 1.27i·13-s + (0.537 + 0.129i)14-s + (0.194 − 0.402i)15-s + (−0.0886 + 0.153i)16-s + (0.852 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.972 - 0.231i$
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: \((2,\ 105,\ (\ :3/2),\ 0.972 - 0.231i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.625528 + 0.0733468i\)
\(L(\frac12)\) \(\approx\) \(0.625528 + 0.0733468i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.18 - 0.383i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (17.7 - 5.23i)T \)
good2 \( 1 + (1.35 + 0.781i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (-26.1 + 15.0i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 59.7iT - 2.19e3T^{2} \)
17 \( 1 + (-59.7 - 103. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-102. - 59.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-65.0 - 37.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 189. iT - 2.43e4T^{2} \)
31 \( 1 + (211. - 122. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (61.8 - 107. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 344.T + 6.89e4T^{2} \)
43 \( 1 - 88.4T + 7.95e4T^{2} \)
47 \( 1 + (278. - 481. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (78.7 - 45.4i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-136. - 236. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-368. - 212. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-60.2 - 104. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 222. iT - 3.57e5T^{2} \)
73 \( 1 + (-853. + 493. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-335. + 580. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 778.T + 5.71e5T^{2} \)
89 \( 1 + (406. - 704. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 288. iT - 9.12e5T^{2} \)
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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

−13.07080246262560962306421609396, −12.12110313513484821343214553762, −10.98810855585029981807253237410, −10.19386216609633737544112287521, −9.395789247896510008822619754295, −7.82348579483738657132573594189, −6.17554575234789140336338847703, −5.52359815506276790088153663643, −3.56889893925143036486528808306, −1.01221655021364186124950284601, 0.65427676957673523939662736477, 3.72315745201460983962821861869, 5.04365989679713496894937165056, 6.86017176534366027508830288120, 7.29545633051829903472689225911, 9.264000356131249522648150944400, 9.593261710894409179534346045248, 11.34103712360999268273822874517, 12.19238876573206178035626411414, 12.97779326262032128813006653202

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