↑

Properties

Label 2-105-105.44-c2-0-18
Degree $2$
Conductor $105$
Sign $0.597 + 0.801i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.530 − 0.919i)2-s + (−2.96 + 0.483i)3-s + (1.43 + 2.48i)4-s + (0.901 − 4.91i)5-s + (−1.12 + 2.97i)6-s + (3.04 − 6.30i)7-s + 7.29·8-s + (8.53 − 2.86i)9-s + (−4.04 − 3.43i)10-s + (9.99 − 5.76i)11-s + (−5.45 − 6.67i)12-s + 5.56i·13-s + (−4.17 − 6.14i)14-s + (−0.290 + 14.9i)15-s + (−1.87 + 3.24i)16-s + (−7.82 − 13.5i)17-s + ⋯
L(s)  = 1  + (0.265 − 0.459i)2-s + (−0.986 + 0.161i)3-s + (0.359 + 0.622i)4-s + (0.180 − 0.983i)5-s + (−0.187 + 0.496i)6-s + (0.435 − 0.900i)7-s + 0.912·8-s + (0.948 − 0.318i)9-s + (−0.404 − 0.343i)10-s + (0.908 − 0.524i)11-s + (−0.454 − 0.555i)12-s + 0.428i·13-s + (−0.298 − 0.439i)14-s + (−0.0193 + 0.999i)15-s + (−0.117 + 0.202i)16-s + (−0.460 − 0.797i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.597 + 0.801i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.597 + 0.801i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24756 - 0.625865i\)
\(L(\frac12)\) \(\approx\) \(1.24756 - 0.625865i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.96 - 0.483i)T \)
5 \( 1 + (-0.901 + 4.91i)T \)
7 \( 1 + (-3.04 + 6.30i)T \)
good2 \( 1 + (-0.530 + 0.919i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (-9.99 + 5.76i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 5.56iT - 169T^{2} \)
17 \( 1 + (7.82 + 13.5i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.06 + 1.84i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (13.0 - 22.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 13.3iT - 841T^{2} \)
31 \( 1 + (-25.3 - 43.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-19.8 - 11.4i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 53.9iT - 1.68e3T^{2} \)
43 \( 1 - 38.8iT - 1.84e3T^{2} \)
47 \( 1 + (2.72 - 4.71i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (15.5 + 26.8i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (97.9 - 56.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (14.8 - 25.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (23.5 - 13.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 79.6iT - 5.04e3T^{2} \)
73 \( 1 + (1.70 - 0.981i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-43.8 + 75.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 71.3T + 6.88e3T^{2} \)
89 \( 1 + (-106. - 61.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 1.02iT - 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.24671973272882658539376204197, −11.97611255592199752823536106098, −11.60563829238976187422780273702, −10.50003293382558076326892445497, −9.185048959912938487050451993531, −7.70458771512267212686758844269, −6.50144573010741809890631306500, −4.85724531781899137546224360926, −3.93955515206484403126615733110, −1.31039483097622593488844790063, 1.96420466559638108415540453247, 4.57929378827655541886262267347, 6.01753740944115514801361611161, 6.44325709288695662724206857525, 7.75937685225043497752060951503, 9.692469078275023998189277361089, 10.69795011249530405961849400594, 11.46323756020399423084868263622, 12.49428940699124724658264499467, 13.90185712049079237408430589940

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