↑

Properties

Label 2-102-51.47-c4-0-0
Degree $2$
Conductor $102$
Sign $-0.467 + 0.884i$
Analytic cond. $10.5437$
Root an. cond. $3.24711$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·2-s + (−0.433 + 8.98i)3-s + 8.00·4-s + (−21.7 + 21.7i)5-s + (1.22 − 25.4i)6-s + (−17.2 + 17.2i)7-s − 22.6·8-s + (−80.6 − 7.79i)9-s + (61.5 − 61.5i)10-s + (43.7 + 43.7i)11-s + (−3.46 + 71.9i)12-s − 22.4·13-s + (48.7 − 48.7i)14-s + (−186. − 204. i)15-s + 64.0·16-s + (281. − 64.4i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.0481 + 0.998i)3-s + 0.500·4-s + (−0.870 + 0.870i)5-s + (0.0340 − 0.706i)6-s + (−0.351 + 0.351i)7-s − 0.353·8-s + (−0.995 − 0.0962i)9-s + (0.615 − 0.615i)10-s + (0.361 + 0.361i)11-s + (−0.0240 + 0.499i)12-s − 0.133·13-s + (0.248 − 0.248i)14-s + (−0.827 − 0.911i)15-s + 0.250·16-s + (0.974 − 0.222i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.884i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $-0.467 + 0.884i$
Analytic conductor: \(10.5437\)
Root analytic conductor: \(3.24711\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{102} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :2),\ -0.467 + 0.884i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0955470 - 0.158586i\)
\(L(\frac12)\) \(\approx\) \(0.0955470 - 0.158586i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82T \)
3 \( 1 + (0.433 - 8.98i)T \)
17 \( 1 + (-281. + 64.4i)T \)
good5 \( 1 + (21.7 - 21.7i)T - 625iT^{2} \)
7 \( 1 + (17.2 - 17.2i)T - 2.40e3iT^{2} \)
11 \( 1 + (-43.7 - 43.7i)T + 1.46e4iT^{2} \)
13 \( 1 + 22.4T + 2.85e4T^{2} \)
19 \( 1 + 35.5iT - 1.30e5T^{2} \)
23 \( 1 + (670. + 670. i)T + 2.79e5iT^{2} \)
29 \( 1 + (-149. + 149. i)T - 7.07e5iT^{2} \)
31 \( 1 + (493. + 493. i)T + 9.23e5iT^{2} \)
37 \( 1 + (981. + 981. i)T + 1.87e6iT^{2} \)
41 \( 1 + (320. + 320. i)T + 2.82e6iT^{2} \)
43 \( 1 - 1.95e3iT - 3.41e6T^{2} \)
47 \( 1 - 699. iT - 4.87e6T^{2} \)
53 \( 1 - 1.29e3T + 7.89e6T^{2} \)
59 \( 1 - 5.80e3T + 1.21e7T^{2} \)
61 \( 1 + (2.97e3 - 2.97e3i)T - 1.38e7iT^{2} \)
67 \( 1 + 1.82e3T + 2.01e7T^{2} \)
71 \( 1 + (3.24e3 - 3.24e3i)T - 2.54e7iT^{2} \)
73 \( 1 + (5.48e3 + 5.48e3i)T + 2.83e7iT^{2} \)
79 \( 1 + (5.69e3 - 5.69e3i)T - 3.89e7iT^{2} \)
83 \( 1 + 1.00e4T + 4.74e7T^{2} \)
89 \( 1 - 2.73e3iT - 6.27e7T^{2} \)
97 \( 1 + (5.20e3 + 5.20e3i)T + 8.85e7iT^{2} \)
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

−14.34152016247905767856430356108, −12.28756552819864806294559305094, −11.48574344120544650377733651239, −10.46430711167952870170516267907, −9.672936022898172615608372030556, −8.474564643575120636455768315372, −7.29211805824660064990315261331, −5.93012591113508178894012121183, −4.08986418279842079923592785974, −2.81660554899157854129907413554, 0.11104680274801169612521752937, 1.41856078014900077253224895561, 3.56373687657316078772321811899, 5.61524677763149773266242039949, 7.03752245541236642445778714413, 7.980916040463093930503271516949, 8.776887810989563900791328786250, 10.17474808244405249171897707380, 11.69290982712497974770740195660, 12.12554825902960868057997037923

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