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Properties

Label 2-105-5.4-c3-0-11
Degree $2$
Conductor $105$
Sign $0.158 + 0.987i$
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  + 0.428i·2-s − 3i·3-s + 7.81·4-s + (−1.76 − 11.0i)5-s + 1.28·6-s − 7i·7-s + 6.77i·8-s − 9·9-s + (4.72 − 0.757i)10-s − 27.4·11-s − 23.4i·12-s − 46.5i·13-s + 2.99·14-s + (−33.1 + 5.30i)15-s + 59.6·16-s − 5.20i·17-s + ⋯
L(s)  = 1  + 0.151i·2-s − 0.577i·3-s + 0.977·4-s + (−0.158 − 0.987i)5-s + 0.0874·6-s − 0.377i·7-s + 0.299i·8-s − 0.333·9-s + (0.149 − 0.0239i)10-s − 0.753·11-s − 0.564i·12-s − 0.993i·13-s + 0.0572·14-s + (−0.570 + 0.0913i)15-s + 0.931·16-s − 0.0742i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.158 + 0.987i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.158 + 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.33411 - 1.13737i\)
\(L(\frac12)\) \(\approx\) \(1.33411 - 1.13737i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 + (1.76 + 11.0i)T \)
7 \( 1 + 7iT \)
good2 \( 1 - 0.428iT - 8T^{2} \)
11 \( 1 + 27.4T + 1.33e3T^{2} \)
13 \( 1 + 46.5iT - 2.19e3T^{2} \)
17 \( 1 + 5.20iT - 4.91e3T^{2} \)
19 \( 1 - 91.0T + 6.85e3T^{2} \)
23 \( 1 + 111. iT - 1.21e4T^{2} \)
29 \( 1 + 0.0763T + 2.43e4T^{2} \)
31 \( 1 - 201.T + 2.97e4T^{2} \)
37 \( 1 - 312. iT - 5.06e4T^{2} \)
41 \( 1 - 102.T + 6.89e4T^{2} \)
43 \( 1 - 257. iT - 7.95e4T^{2} \)
47 \( 1 - 350. iT - 1.03e5T^{2} \)
53 \( 1 - 196. iT - 1.48e5T^{2} \)
59 \( 1 + 881.T + 2.05e5T^{2} \)
61 \( 1 - 737.T + 2.26e5T^{2} \)
67 \( 1 + 365. iT - 3.00e5T^{2} \)
71 \( 1 - 1.11e3T + 3.57e5T^{2} \)
73 \( 1 - 261. iT - 3.89e5T^{2} \)
79 \( 1 + 273.T + 4.93e5T^{2} \)
83 \( 1 + 87.1iT - 5.71e5T^{2} \)
89 \( 1 - 1.09e3T + 7.04e5T^{2} \)
97 \( 1 + 228. iT - 9.12e5T^{2} \)
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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

−12.88577672339804314219225100269, −12.15410419008600437938056181899, −11.07807238437069614113528286092, −9.918430751862060733641999654867, −8.221858903409402363274938119492, −7.63260051473435191132272506511, −6.23923352954947796229789468733, −5.00718985857788251436213541895, −2.87180820998379020100398082014, −1.01861256683466620145631022893, 2.34324822840924792785866273889, 3.58427089350322792791823089277, 5.52797727009820692301037337687, 6.78950790966374679914279276152, 7.82522180223221112132653262826, 9.499475118200740161686517061087, 10.47981240725889895473708313187, 11.36345180390108925763177355713, 12.05469494883976185694999577586, 13.69848464714311473104675022402

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