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Properties

Label 2-105-35.27-c3-0-10
Degree $2$
Conductor $105$
Sign $0.837 - 0.546i$
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  + (1.52 + 1.52i)2-s + (−2.12 − 2.12i)3-s − 3.32i·4-s + (−1.55 + 11.0i)5-s − 6.48i·6-s + (15.0 + 10.7i)7-s + (17.3 − 17.3i)8-s + 8.99i·9-s + (−19.3 + 14.5i)10-s + 55.7·11-s + (−7.05 + 7.05i)12-s + (29.8 + 29.8i)13-s + (6.56 + 39.5i)14-s + (26.7 − 20.1i)15-s + 26.3·16-s + (65.5 − 65.5i)17-s + ⋯
L(s)  = 1  + (0.540 + 0.540i)2-s + (−0.408 − 0.408i)3-s − 0.415i·4-s + (−0.139 + 0.990i)5-s − 0.441i·6-s + (0.813 + 0.581i)7-s + (0.765 − 0.765i)8-s + 0.333i·9-s + (−0.610 + 0.459i)10-s + 1.52·11-s + (−0.169 + 0.169i)12-s + (0.636 + 0.636i)13-s + (0.125 + 0.754i)14-s + (0.461 − 0.347i)15-s + 0.411·16-s + (0.935 − 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.837 - 0.546i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.837 - 0.546i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.99727 + 0.594373i\)
\(L(\frac12)\) \(\approx\) \(1.99727 + 0.594373i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.55 - 11.0i)T \)
7 \( 1 + (-15.0 - 10.7i)T \)
good2 \( 1 + (-1.52 - 1.52i)T + 8iT^{2} \)
11 \( 1 - 55.7T + 1.33e3T^{2} \)
13 \( 1 + (-29.8 - 29.8i)T + 2.19e3iT^{2} \)
17 \( 1 + (-65.5 + 65.5i)T - 4.91e3iT^{2} \)
19 \( 1 + 85.1T + 6.85e3T^{2} \)
23 \( 1 + (75.8 - 75.8i)T - 1.21e4iT^{2} \)
29 \( 1 - 44.1iT - 2.43e4T^{2} \)
31 \( 1 - 192. iT - 2.97e4T^{2} \)
37 \( 1 + (139. + 139. i)T + 5.06e4iT^{2} \)
41 \( 1 + 251. iT - 6.89e4T^{2} \)
43 \( 1 + (-19.2 + 19.2i)T - 7.95e4iT^{2} \)
47 \( 1 + (-321. + 321. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-43.4 + 43.4i)T - 1.48e5iT^{2} \)
59 \( 1 + 97.4T + 2.05e5T^{2} \)
61 \( 1 - 60.2iT - 2.26e5T^{2} \)
67 \( 1 + (466. + 466. i)T + 3.00e5iT^{2} \)
71 \( 1 + 522.T + 3.57e5T^{2} \)
73 \( 1 + (-141. - 141. i)T + 3.89e5iT^{2} \)
79 \( 1 - 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 + (1.04e3 + 1.04e3i)T + 5.71e5iT^{2} \)
89 \( 1 - 168.T + 7.04e5T^{2} \)
97 \( 1 + (-133. + 133. i)T - 9.12e5iT^{2} \)
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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

−13.99603709454061068877102930395, −12.18283097563199074453935282063, −11.43603878902724123872247154111, −10.41158121771868112916692496381, −8.968938531900920783555692809300, −7.33714754145228724469401580166, −6.49822527651355410551292325848, −5.52319889855145481577560825409, −3.99111962332971277772178110609, −1.64242916420506270150359899530, 1.37537459454456779170745935112, 3.88687774418915369618482852954, 4.45693234382916688388433262565, 5.98575806461908638048458779192, 7.890304519864065550059835831219, 8.716139718894838104911299960571, 10.29060473683733049826648810688, 11.35094891971647436804978544374, 12.10885568649784803168344723588, 12.94908286652244448135392180750

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