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Properties

Label 2-105-35.18-c2-0-9
Degree $2$
Conductor $105$
Sign $0.915 - 0.403i$
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  + (2.56 + 0.686i)2-s + (1.67 − 0.448i)3-s + (2.62 + 1.51i)4-s + (1.32 + 4.81i)5-s + 4.59·6-s + (0.158 − 6.99i)7-s + (−1.82 − 1.82i)8-s + (2.59 − 1.50i)9-s + (0.0981 + 13.2i)10-s + (−8.05 + 13.9i)11-s + (5.06 + 1.35i)12-s + (−0.148 − 0.148i)13-s + (5.20 − 17.8i)14-s + (4.38 + 7.46i)15-s + (−9.47 − 16.4i)16-s + (−5.20 − 19.4i)17-s + ⋯
L(s)  = 1  + (1.28 + 0.343i)2-s + (0.557 − 0.149i)3-s + (0.655 + 0.378i)4-s + (0.265 + 0.963i)5-s + 0.765·6-s + (0.0227 − 0.999i)7-s + (−0.227 − 0.227i)8-s + (0.288 − 0.166i)9-s + (0.00981 + 1.32i)10-s + (−0.732 + 1.26i)11-s + (0.422 + 0.113i)12-s + (−0.0114 − 0.0114i)13-s + (0.372 − 1.27i)14-s + (0.292 + 0.497i)15-s + (−0.592 − 1.02i)16-s + (−0.306 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.915 - 0.403i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.915 - 0.403i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.68871 + 0.565981i\)
\(L(\frac12)\) \(\approx\) \(2.68871 + 0.565981i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 + 0.448i)T \)
5 \( 1 + (-1.32 - 4.81i)T \)
7 \( 1 + (-0.158 + 6.99i)T \)
good2 \( 1 + (-2.56 - 0.686i)T + (3.46 + 2i)T^{2} \)
11 \( 1 + (8.05 - 13.9i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (0.148 + 0.148i)T + 169iT^{2} \)
17 \( 1 + (5.20 + 19.4i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-10.9 + 6.34i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (4.18 - 15.6i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 21.7iT - 841T^{2} \)
31 \( 1 + (-7.17 + 12.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (62.8 + 16.8i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 2.26T + 1.68e3T^{2} \)
43 \( 1 + (-35.1 - 35.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-55.8 - 14.9i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-101. + 27.1i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-59.5 - 34.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-17.3 - 30.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-8.56 - 31.9i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 59.5T + 5.04e3T^{2} \)
73 \( 1 + (-33.4 + 8.96i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-2.87 + 1.65i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-4.48 - 4.48i)T + 6.88e3iT^{2} \)
89 \( 1 + (123. - 71.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-113. + 113. i)T - 9.40e3iT^{2} \)
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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

−13.73709862355139200147744462079, −13.04504305185924413570681886339, −11.79450555623191581811932404229, −10.40946387067803993438819908083, −9.471441447554172646773501146701, −7.30298304641321490423992631060, −7.05266792571525234817555551908, −5.32124749353793488031267959353, −4.01252399084993605467640954993, −2.68498749607496045428889689772, 2.35428928339106949965242621736, 3.78559758248029575086391671830, 5.20610258265742845698603192364, 5.96162281204676136218783750954, 8.362362020786546799916029811098, 8.821111975027038601080001874692, 10.44815974093213007506119736261, 11.83662417656022782043621751604, 12.59823438331129271500945970959, 13.45175046275193394086152753333

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