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Properties

Label 2-105-35.13-c3-0-23
Degree $2$
Conductor $105$
Sign $-0.990 + 0.140i$
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  + (3.82 − 3.82i)2-s + (−2.12 + 2.12i)3-s − 21.3i·4-s + (−10.1 − 4.60i)5-s + 16.2i·6-s + (−17.4 + 6.06i)7-s + (−51.0 − 51.0i)8-s − 8.99i·9-s + (−56.6 + 21.3i)10-s + 38.1·11-s + (45.2 + 45.2i)12-s + (62.7 − 62.7i)13-s + (−43.7 + 90.2i)14-s + (31.3 − 11.8i)15-s − 220.·16-s + (−15.5 − 15.5i)17-s + ⋯
L(s)  = 1  + (1.35 − 1.35i)2-s + (−0.408 + 0.408i)3-s − 2.66i·4-s + (−0.911 − 0.411i)5-s + 1.10i·6-s + (−0.944 + 0.327i)7-s + (−2.25 − 2.25i)8-s − 0.333i·9-s + (−1.79 + 0.676i)10-s + 1.04·11-s + (1.08 + 1.08i)12-s + (1.33 − 1.33i)13-s + (−0.835 + 1.72i)14-s + (0.540 − 0.204i)15-s − 3.43·16-s + (−0.221 − 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.990 + 0.140i$
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: \((2,\ 105,\ (\ :3/2),\ -0.990 + 0.140i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.142085 - 2.00814i\)
\(L(\frac12)\) \(\approx\) \(0.142085 - 2.00814i\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.1 + 4.60i)T \)
7 \( 1 + (17.4 - 6.06i)T \)
good2 \( 1 + (-3.82 + 3.82i)T - 8iT^{2} \)
11 \( 1 - 38.1T + 1.33e3T^{2} \)
13 \( 1 + (-62.7 + 62.7i)T - 2.19e3iT^{2} \)
17 \( 1 + (15.5 + 15.5i)T + 4.91e3iT^{2} \)
19 \( 1 - 25.6T + 6.85e3T^{2} \)
23 \( 1 + (21.8 + 21.8i)T + 1.21e4iT^{2} \)
29 \( 1 + 82.7iT - 2.43e4T^{2} \)
31 \( 1 - 197. iT - 2.97e4T^{2} \)
37 \( 1 + (-111. + 111. i)T - 5.06e4iT^{2} \)
41 \( 1 + 28.5iT - 6.89e4T^{2} \)
43 \( 1 + (-76.5 - 76.5i)T + 7.95e4iT^{2} \)
47 \( 1 + (265. + 265. i)T + 1.03e5iT^{2} \)
53 \( 1 + (124. + 124. i)T + 1.48e5iT^{2} \)
59 \( 1 - 480.T + 2.05e5T^{2} \)
61 \( 1 - 246. iT - 2.26e5T^{2} \)
67 \( 1 + (179. - 179. i)T - 3.00e5iT^{2} \)
71 \( 1 - 142.T + 3.57e5T^{2} \)
73 \( 1 + (206. - 206. i)T - 3.89e5iT^{2} \)
79 \( 1 - 982. iT - 4.93e5T^{2} \)
83 \( 1 + (-433. + 433. i)T - 5.71e5iT^{2} \)
89 \( 1 - 864.T + 7.04e5T^{2} \)
97 \( 1 + (-814. - 814. i)T + 9.12e5iT^{2} \)
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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

−12.63739085324919077516604890888, −11.83637609425755930069581642120, −11.05948686854490691778888918485, −10.03055734285196341483875110868, −8.856681981699132259854628848593, −6.44301847735273855632050869163, −5.38566384495105433168444467345, −4.00570549972554329851162022844, −3.24898484404073297799629401499, −0.814720618307578980136920488891, 3.51510596144922725480714032340, 4.31789062133313303754166154812, 6.23907302554613554088854977512, 6.62093515548209680163858899582, 7.67947481027147566927263585130, 8.980736737842541368581753570165, 11.28808553702186043188751799737, 11.95773748415909982194344340557, 13.03821930164616925245853950565, 13.84153773756992363423815388552

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