↑

Properties

Label 2-170-17.8-c1-0-0
Degree $2$
Conductor $170$
Sign $-0.916 + 0.399i$
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 + (−0.894 + 2.15i)3-s − 1.00i·4-s + (−0.923 − 0.382i)5-s + (−0.894 − 2.15i)6-s + (−3.32 + 1.37i)7-s + (0.707 + 0.707i)8-s + (−1.74 − 1.74i)9-s + (0.923 − 0.382i)10-s + (−0.760 − 1.83i)11-s + (2.15 + 0.894i)12-s + 0.678i·13-s + (1.37 − 3.32i)14-s + (1.65 − 1.65i)15-s − 1.00·16-s + (−2.74 − 3.07i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.516 + 1.24i)3-s − 0.500i·4-s + (−0.413 − 0.171i)5-s + (−0.365 − 0.881i)6-s + (−1.25 + 0.520i)7-s + (0.250 + 0.250i)8-s + (−0.580 − 0.580i)9-s + (0.292 − 0.121i)10-s + (−0.229 − 0.553i)11-s + (0.623 + 0.258i)12-s + 0.188i·13-s + (0.367 − 0.888i)14-s + (0.426 − 0.426i)15-s − 0.250·16-s + (−0.665 − 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.916 + 0.399i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.916 + 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0673796 - 0.322974i\)
\(L(\frac12)\) \(\approx\) \(0.0673796 - 0.322974i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 + (2.74 + 3.07i)T \)
good3 \( 1 + (0.894 - 2.15i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (3.32 - 1.37i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.760 + 1.83i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 0.678iT - 13T^{2} \)
19 \( 1 + (3.09 - 3.09i)T - 19iT^{2} \)
23 \( 1 + (-1.51 - 3.65i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-5.47 - 2.26i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (3.63 - 8.76i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (1.64 - 3.98i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.70 - 0.707i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-8.51 - 8.51i)T + 43iT^{2} \)
47 \( 1 + 13.6iT - 47T^{2} \)
53 \( 1 + (4.30 - 4.30i)T - 53iT^{2} \)
59 \( 1 + (0.760 + 0.760i)T + 59iT^{2} \)
61 \( 1 + (-1.68 + 0.698i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 5.62T + 67T^{2} \)
71 \( 1 + (4.51 - 10.9i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.67 + 2.34i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (1.60 + 3.88i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-11.8 + 11.8i)T - 83iT^{2} \)
89 \( 1 - 0.600iT - 89T^{2} \)
97 \( 1 + (-0.119 - 0.0496i)T + (68.5 + 68.5i)T^{2} \)
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

−13.33868179548151667056442878063, −12.16064854874372519090658004545, −11.05321401834072576930093657402, −10.21146035651944722099812723072, −9.328251501433889896382551022245, −8.555204817489469371674537062089, −6.94500944066945689763292494521, −5.84789761992503739579252990711, −4.76522087924696278781150139951, −3.31893282770701065746236536600, 0.36679034321025497249872151997, 2.41436003045603191358401657485, 4.11267168466492217146757933814, 6.25328058116135759860042278613, 6.96427851514671194637073277894, 7.88747735741240536741888031772, 9.216059230032314464656811069523, 10.41610203058733555554352379749, 11.20842099811767588184241613023, 12.51913889062194062929153102010

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