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Properties

Label 2-105-105.104-c1-0-1
Degree $2$
Conductor $105$
Sign $-0.645 - 0.763i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 + 1.11i)3-s − 2·4-s + 2.23i·5-s − 2.64·7-s + (0.5 − 2.95i)9-s + 5.91i·11-s + (2.64 − 2.23i)12-s + 2.64·13-s + (−2.50 − 2.95i)15-s + 4·16-s − 2.23i·17-s − 4.47i·20-s + (3.50 − 2.95i)21-s − 5.00·25-s + (2.64 + 4.47i)27-s + 5.29·28-s + ⋯
L(s)  = 1  + (−0.763 + 0.645i)3-s − 4-s + 0.999i·5-s − 0.999·7-s + (0.166 − 0.986i)9-s + 1.78i·11-s + (0.763 − 0.645i)12-s + 0.733·13-s + (−0.645 − 0.763i)15-s + 16-s − 0.542i·17-s − 0.999i·20-s + (0.763 − 0.645i)21-s − 1.00·25-s + (0.509 + 0.860i)27-s + 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.645 - 0.763i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.645 - 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210750 + 0.454054i\)
\(L(\frac12)\) \(\approx\) \(0.210750 + 0.454054i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.32 - 1.11i)T \)
5 \( 1 - 2.23iT \)
7 \( 1 + 2.64T \)
good2 \( 1 + 2T^{2} \)
11 \( 1 - 5.91iT - 11T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 5.91iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + 8.94iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 18.5T + 97T^{2} \)
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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

−14.25893519281782353489141038097, −12.99168797998621854376252451316, −12.13744092956451481461377199722, −10.73282313894775253261756710319, −9.897870467817826874600174401912, −9.224795699877389279750112784389, −7.26360276363798949662648662762, −6.15362876635385216636892125316, −4.71280274439209525804059574563, −3.47110627331364208458537828722, 0.66368167937209263439652868992, 3.82719619637920896474875253525, 5.45555262408562740704452774358, 6.21577155880049371115133005822, 8.123299798078636550615703189610, 8.870322178775433819924699679718, 10.17388558710500914193221020720, 11.47685159915176114164733461736, 12.60586758942157858160174748866, 13.34741294708740758107369255322

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