↑

Properties

Label 2-170-85.9-c1-0-8
Degree $2$
Conductor $170$
Sign $0.933 + 0.359i$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + (1.23 − 2.99i)3-s + 1.00i·4-s + (1.87 + 1.22i)5-s + (2.99 − 1.23i)6-s + (−1.49 + 0.620i)7-s + (−0.707 + 0.707i)8-s + (−5.30 − 5.30i)9-s + (0.458 + 2.18i)10-s + (−4.11 + 1.70i)11-s + (2.99 + 1.23i)12-s + 1.78·13-s + (−1.49 − 0.620i)14-s + (5.98 − 4.08i)15-s − 1.00·16-s + (3.35 + 2.39i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.715 − 1.72i)3-s + 0.500i·4-s + (0.837 + 0.547i)5-s + (1.22 − 0.506i)6-s + (−0.566 + 0.234i)7-s + (−0.250 + 0.250i)8-s + (−1.76 − 1.76i)9-s + (0.144 + 0.692i)10-s + (−1.24 + 0.514i)11-s + (0.864 + 0.357i)12-s + 0.496·13-s + (−0.400 − 0.165i)14-s + (1.54 − 1.05i)15-s − 0.250·16-s + (0.813 + 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.933 + 0.359i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.933 + 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74743 - 0.324805i\)
\(L(\frac12)\) \(\approx\) \(1.74743 - 0.324805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-1.87 - 1.22i)T \)
17 \( 1 + (-3.35 - 2.39i)T \)
good3 \( 1 + (-1.23 + 2.99i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (1.49 - 0.620i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (4.11 - 1.70i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 1.78T + 13T^{2} \)
19 \( 1 + (0.157 - 0.157i)T - 19iT^{2} \)
23 \( 1 + (2.07 + 4.99i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (2.00 - 4.83i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (1.60 + 0.666i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-3.91 + 9.44i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.657 - 1.58i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (3.65 - 3.65i)T - 43iT^{2} \)
47 \( 1 - 5.48T + 47T^{2} \)
53 \( 1 + (-1.42 - 1.42i)T + 53iT^{2} \)
59 \( 1 + (-1.74 - 1.74i)T + 59iT^{2} \)
61 \( 1 + (3.62 + 8.74i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 14.4iT - 67T^{2} \)
71 \( 1 + (-7.59 - 3.14i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.40 + 0.582i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (2.76 - 1.14i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-10.0 - 10.0i)T + 83iT^{2} \)
89 \( 1 + 9.77iT - 89T^{2} \)
97 \( 1 + (4.59 + 1.90i)T + (68.5 + 68.5i)T^{2} \)
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

−12.67543581710763965967429460985, −12.59072245154513446996233213630, −10.83840598056627226610639416047, −9.418622120243714297709882825493, −8.213972239829653401308578997517, −7.38381521077809337224398917789, −6.42246738742264196239615710204, −5.67270093101443269695985066366, −3.18810720432419047129749972497, −2.13345392645085196203430724212, 2.69540286121399407171316941197, 3.74570031033573513073129765367, 5.05454924253165674915726200849, 5.77301121249407201378677825375, 8.086424809673863590508857685297, 9.198836056810099902750913338705, 9.969753111569364137948466404994, 10.46831552395131937940458516881, 11.66206825893745700250071817758, 13.38428074351614169382349168486

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