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Properties

Label 2-105-3.2-c2-0-13
Degree $2$
Conductor $105$
Sign $-0.997 + 0.0723i$
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  − 2.62i·2-s + (−0.217 − 2.99i)3-s − 2.90·4-s − 2.23i·5-s + (−7.86 + 0.570i)6-s + 2.64·7-s − 2.87i·8-s + (−8.90 + 1.29i)9-s − 5.87·10-s + 9.48i·11-s + (0.630 + 8.69i)12-s + 9.27·13-s − 6.95i·14-s + (−6.69 + 0.485i)15-s − 19.1·16-s + 11.2i·17-s + ⋯
L(s)  = 1  − 1.31i·2-s + (−0.0723 − 0.997i)3-s − 0.726·4-s − 0.447i·5-s + (−1.31 + 0.0950i)6-s + 0.377·7-s − 0.359i·8-s + (−0.989 + 0.144i)9-s − 0.587·10-s + 0.862i·11-s + (0.0525 + 0.724i)12-s + 0.713·13-s − 0.496i·14-s + (−0.446 + 0.0323i)15-s − 1.19·16-s + 0.663i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0723i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.997 + 0.0723i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.997 + 0.0723i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0480123 - 1.32578i\)
\(L(\frac12)\) \(\approx\) \(0.0480123 - 1.32578i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.217 + 2.99i)T \)
5 \( 1 + 2.23iT \)
7 \( 1 - 2.64T \)
good2 \( 1 + 2.62iT - 4T^{2} \)
11 \( 1 - 9.48iT - 121T^{2} \)
13 \( 1 - 9.27T + 169T^{2} \)
17 \( 1 - 11.2iT - 289T^{2} \)
19 \( 1 - 18.4T + 361T^{2} \)
23 \( 1 + 33.2iT - 529T^{2} \)
29 \( 1 + 48.1iT - 841T^{2} \)
31 \( 1 + 8.45T + 961T^{2} \)
37 \( 1 - 47.5T + 1.36e3T^{2} \)
41 \( 1 - 11.7iT - 1.68e3T^{2} \)
43 \( 1 - 14.6T + 1.84e3T^{2} \)
47 \( 1 - 34.3iT - 2.20e3T^{2} \)
53 \( 1 - 13.8iT - 2.80e3T^{2} \)
59 \( 1 - 99.0iT - 3.48e3T^{2} \)
61 \( 1 - 72.7T + 3.72e3T^{2} \)
67 \( 1 + 129.T + 4.48e3T^{2} \)
71 \( 1 - 121. iT - 5.04e3T^{2} \)
73 \( 1 - 61.1T + 5.32e3T^{2} \)
79 \( 1 + 112.T + 6.24e3T^{2} \)
83 \( 1 + 102. iT - 6.88e3T^{2} \)
89 \( 1 - 15.3iT - 7.92e3T^{2} \)
97 \( 1 - 84.1T + 9.40e3T^{2} \)
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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

−12.79945278472060909991454412975, −11.97729541124676008592634564383, −11.21757557335407323459248960957, −10.03869264636738371371556437066, −8.761788524901788772390499109472, −7.54727170605639373693426034911, −6.11070720160801291731743525894, −4.31089596268741222977147088673, −2.47413345085552329519932403615, −1.12594906099648315506314434639, 3.36940639378451522858944208125, 5.10664624766465733965290670464, 5.95222449497978957955765067849, 7.33260881801913980177307048480, 8.481491194803221526096507396289, 9.467458714226502871978853379119, 10.97350047344271705794547507811, 11.52938602189817600606210950199, 13.65849040389849239256104720167, 14.29062457476300618092909741927

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