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Properties

Label 2-105-15.14-c2-0-19
Degree $2$
Conductor $105$
Sign $0.802 + 0.596i$
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  + 3.57·2-s + (−2.50 − 1.65i)3-s + 8.75·4-s + (0.280 − 4.99i)5-s + (−8.94 − 5.90i)6-s + 2.64i·7-s + 16.9·8-s + (3.53 + 8.27i)9-s + (1.00 − 17.8i)10-s − 2.85i·11-s + (−21.9 − 14.4i)12-s + 20.2i·13-s + 9.44i·14-s + (−8.95 + 12.0i)15-s + 25.6·16-s − 25.2·17-s + ⋯
L(s)  = 1  + 1.78·2-s + (−0.834 − 0.550i)3-s + 2.18·4-s + (0.0560 − 0.998i)5-s + (−1.49 − 0.983i)6-s + 0.377i·7-s + 2.12·8-s + (0.393 + 0.919i)9-s + (0.100 − 1.78i)10-s − 0.259i·11-s + (−1.82 − 1.20i)12-s + 1.55i·13-s + 0.674i·14-s + (−0.596 + 0.802i)15-s + 1.60·16-s − 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.802 + 0.596i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.802 + 0.596i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.60355 - 0.862036i\)
\(L(\frac12)\) \(\approx\) \(2.60355 - 0.862036i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.50 + 1.65i)T \)
5 \( 1 + (-0.280 + 4.99i)T \)
7 \( 1 - 2.64iT \)
good2 \( 1 - 3.57T + 4T^{2} \)
11 \( 1 + 2.85iT - 121T^{2} \)
13 \( 1 - 20.2iT - 169T^{2} \)
17 \( 1 + 25.2T + 289T^{2} \)
19 \( 1 - 23.2T + 361T^{2} \)
23 \( 1 + 6.18T + 529T^{2} \)
29 \( 1 - 10.2iT - 841T^{2} \)
31 \( 1 + 0.392T + 961T^{2} \)
37 \( 1 - 10.7iT - 1.36e3T^{2} \)
41 \( 1 + 46.2iT - 1.68e3T^{2} \)
43 \( 1 + 35.4iT - 1.84e3T^{2} \)
47 \( 1 - 55.3T + 2.20e3T^{2} \)
53 \( 1 - 19.7T + 2.80e3T^{2} \)
59 \( 1 - 71.2iT - 3.48e3T^{2} \)
61 \( 1 + 58.9T + 3.72e3T^{2} \)
67 \( 1 + 95.3iT - 4.48e3T^{2} \)
71 \( 1 + 100. iT - 5.04e3T^{2} \)
73 \( 1 - 1.93iT - 5.32e3T^{2} \)
79 \( 1 - 40.7T + 6.24e3T^{2} \)
83 \( 1 + 52.6T + 6.88e3T^{2} \)
89 \( 1 - 60.7iT - 7.92e3T^{2} \)
97 \( 1 + 37.6iT - 9.40e3T^{2} \)
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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

−13.57552008697742070346306425170, −12.29297550701599048574191153532, −11.91174936692510070881344386165, −10.96431252071945534310956931776, −9.025915430086118548667735583885, −7.22536853674918444857180519448, −6.17380481315106446954109194447, −5.17298043368573444002914567749, −4.24111147765340462251636824739, −1.97972048230387780382701049062, 2.95712946618328035153134413320, 4.17859306487401734257787749855, 5.41777323880036105360141398776, 6.37558617536018408848015282053, 7.40151145694097899634158960049, 9.966288348792587676411266556458, 10.91578157602823260669255631386, 11.55405371409375603243554355179, 12.71574853741556530335091337291, 13.59338210333999871848981621941

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