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Properties

Label 2-105-105.74-c2-0-24
Degree $2$
Conductor $105$
Sign $-0.995 + 0.0908i$
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  + (−1.48 − 2.57i)2-s + (1.18 − 2.75i)3-s + (−2.43 + 4.22i)4-s + (3.23 − 3.80i)5-s + (−8.87 + 1.06i)6-s + (5.16 − 4.72i)7-s + 2.60·8-s + (−6.21 − 6.51i)9-s + (−14.6 − 2.68i)10-s + (3.56 + 2.05i)11-s + (8.76 + 11.7i)12-s + 21.3i·13-s + (−19.8 − 6.30i)14-s + (−6.67 − 13.4i)15-s + (5.87 + 10.1i)16-s + (−1.11 + 1.92i)17-s + ⋯
L(s)  = 1  + (−0.744 − 1.28i)2-s + (0.393 − 0.919i)3-s + (−0.609 + 1.05i)4-s + (0.647 − 0.761i)5-s + (−1.47 + 0.176i)6-s + (0.738 − 0.674i)7-s + 0.325·8-s + (−0.690 − 0.723i)9-s + (−1.46 − 0.268i)10-s + (0.323 + 0.186i)11-s + (0.730 + 0.975i)12-s + 1.64i·13-s + (−1.41 − 0.450i)14-s + (−0.445 − 0.895i)15-s + (0.367 + 0.635i)16-s + (−0.0653 + 0.113i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0908i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.995 + 0.0908i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.995 + 0.0908i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0518904 - 1.13948i\)
\(L(\frac12)\) \(\approx\) \(0.0518904 - 1.13948i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.18 + 2.75i)T \)
5 \( 1 + (-3.23 + 3.80i)T \)
7 \( 1 + (-5.16 + 4.72i)T \)
good2 \( 1 + (1.48 + 2.57i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (-3.56 - 2.05i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 21.3iT - 169T^{2} \)
17 \( 1 + (1.11 - 1.92i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-4.93 - 8.55i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (9.66 + 16.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 28.1iT - 841T^{2} \)
31 \( 1 + (-17.6 + 30.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-43.1 + 24.9i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 27.4iT - 1.68e3T^{2} \)
43 \( 1 - 41.9iT - 1.84e3T^{2} \)
47 \( 1 + (-19.1 - 33.2i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (25.6 - 44.4i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (22.0 + 12.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (34.1 + 59.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-59.1 - 34.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 11.9iT - 5.04e3T^{2} \)
73 \( 1 + (88.6 + 51.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-32.1 - 55.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 122.T + 6.88e3T^{2} \)
89 \( 1 + (116. - 67.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 13.6iT - 9.40e3T^{2} \)
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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

−12.71689826608742135378780622002, −11.94250645481440231414314660165, −11.02323627907760955705111989131, −9.639733379280096770405190840022, −8.923702187855605920211719955212, −7.86505545594025237726275066162, −6.31817898138447284050839006441, −4.21838962604927604741199973367, −2.15285452680164303372314334841, −1.18291282778892749501470561703, 2.93106739520428817967027018980, 5.21214245427719241148173530381, 6.05965397058580468844998127187, 7.64601609501665248199071735757, 8.496059418621094033095261776282, 9.540538735255650760804338577522, 10.38471660148772282607581433005, 11.61989767861221298591911879830, 13.56887088055867753175093409216, 14.53450347137685401429978577380

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