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Properties

Label 2-105-35.13-c3-0-13
Degree $2$
Conductor $105$
Sign $0.961 + 0.273i$
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  + (−2.16 + 2.16i)2-s + (−2.12 + 2.12i)3-s − 1.37i·4-s + (9.30 − 6.19i)5-s − 9.18i·6-s + (−8.45 − 16.4i)7-s + (−14.3 − 14.3i)8-s − 8.99i·9-s + (−6.72 + 33.5i)10-s − 21.5·11-s + (2.90 + 2.90i)12-s + (48.3 − 48.3i)13-s + (53.9 + 17.3i)14-s + (−6.58 + 32.8i)15-s + 73.0·16-s + (30.9 + 30.9i)17-s + ⋯
L(s)  = 1  + (−0.765 + 0.765i)2-s + (−0.408 + 0.408i)3-s − 0.171i·4-s + (0.832 − 0.554i)5-s − 0.624i·6-s + (−0.456 − 0.889i)7-s + (−0.634 − 0.634i)8-s − 0.333i·9-s + (−0.212 + 1.06i)10-s − 0.590·11-s + (0.0699 + 0.0699i)12-s + (1.03 − 1.03i)13-s + (1.03 + 0.331i)14-s + (−0.113 + 0.566i)15-s + 1.14·16-s + (0.441 + 0.441i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.961 + 0.273i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.961 + 0.273i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.816031 - 0.113734i\)
\(L(\frac12)\) \(\approx\) \(0.816031 - 0.113734i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.12 - 2.12i)T \)
5 \( 1 + (-9.30 + 6.19i)T \)
7 \( 1 + (8.45 + 16.4i)T \)
good2 \( 1 + (2.16 - 2.16i)T - 8iT^{2} \)
11 \( 1 + 21.5T + 1.33e3T^{2} \)
13 \( 1 + (-48.3 + 48.3i)T - 2.19e3iT^{2} \)
17 \( 1 + (-30.9 - 30.9i)T + 4.91e3iT^{2} \)
19 \( 1 - 91.2T + 6.85e3T^{2} \)
23 \( 1 + (143. + 143. i)T + 1.21e4iT^{2} \)
29 \( 1 - 284. iT - 2.43e4T^{2} \)
31 \( 1 + 72.5iT - 2.97e4T^{2} \)
37 \( 1 + (-160. + 160. i)T - 5.06e4iT^{2} \)
41 \( 1 + 404. iT - 6.89e4T^{2} \)
43 \( 1 + (73.2 + 73.2i)T + 7.95e4iT^{2} \)
47 \( 1 + (261. + 261. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-24.1 - 24.1i)T + 1.48e5iT^{2} \)
59 \( 1 - 157.T + 2.05e5T^{2} \)
61 \( 1 + 557. iT - 2.26e5T^{2} \)
67 \( 1 + (206. - 206. i)T - 3.00e5iT^{2} \)
71 \( 1 - 690.T + 3.57e5T^{2} \)
73 \( 1 + (159. - 159. i)T - 3.89e5iT^{2} \)
79 \( 1 + 998. iT - 4.93e5T^{2} \)
83 \( 1 + (66.4 - 66.4i)T - 5.71e5iT^{2} \)
89 \( 1 + 327.T + 7.04e5T^{2} \)
97 \( 1 + (-890. - 890. i)T + 9.12e5iT^{2} \)
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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

−13.16037896317045178813852657326, −12.41550811913077441916326276392, −10.56927250730905042080378748701, −10.00183631971348286288282293749, −8.826934358885947733520339489602, −7.78981375660086684766242017729, −6.43634086439751201112852422820, −5.44590041767907280585375270379, −3.57787282113162710919688506514, −0.65285087423150517721039523814, 1.57169248470760207214678722853, 2.86813952566261376137991604908, 5.58359593101513825713725635776, 6.31180870338968265672455463467, 8.034892907222807881942811300094, 9.500689588352284761474298808253, 9.903434623645553841931352393893, 11.36731355414325213275201215863, 11.77289755350224892676607300532, 13.28052640752823983205032285191

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