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Properties

Label 2-170-85.62-c1-0-8
Degree $2$
Conductor $170$
Sign $-0.768 + 0.639i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 + 0.923i)2-s + (−1.56 − 1.04i)3-s + (−0.707 + 0.707i)4-s + (−2.19 − 0.435i)5-s + (0.366 − 1.84i)6-s + (−3.19 − 0.635i)7-s + (−0.923 − 0.382i)8-s + (0.199 + 0.481i)9-s + (−0.436 − 2.19i)10-s + (−0.447 − 0.0890i)11-s + (1.84 − 0.366i)12-s + 0.527i·13-s + (−0.635 − 3.19i)14-s + (2.96 + 2.96i)15-s − i·16-s + (−2.29 + 3.42i)17-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (−0.900 − 0.601i)3-s + (−0.353 + 0.353i)4-s + (−0.980 − 0.194i)5-s + (0.149 − 0.751i)6-s + (−1.20 − 0.240i)7-s + (−0.326 − 0.135i)8-s + (0.0664 + 0.160i)9-s + (−0.138 − 0.693i)10-s + (−0.134 − 0.0268i)11-s + (0.531 − 0.105i)12-s + 0.146i·13-s + (−0.169 − 0.854i)14-s + (0.766 + 0.765i)15-s − 0.250i·16-s + (−0.555 + 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.768 + 0.639i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.768 + 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0554465 - 0.153401i\)
\(L(\frac12)\) \(\approx\) \(0.0554465 - 0.153401i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 0.923i)T \)
5 \( 1 + (2.19 + 0.435i)T \)
17 \( 1 + (2.29 - 3.42i)T \)
good3 \( 1 + (1.56 + 1.04i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (3.19 + 0.635i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (0.447 + 0.0890i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 - 0.527iT - 13T^{2} \)
19 \( 1 + (-1.16 + 2.80i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.17 + 6.25i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (-5.65 - 3.78i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (3.44 - 0.684i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (-4.60 + 6.88i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (6.26 - 4.18i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-0.617 + 1.48i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 12.7T + 47T^{2} \)
53 \( 1 + (4.19 - 1.73i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.43 + 1.42i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (5.84 + 8.74i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (-3.29 + 3.29i)T - 67iT^{2} \)
71 \( 1 + (2.03 + 10.2i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-7.39 + 1.47i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-0.0264 + 0.132i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (3.81 + 9.21i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (5.04 - 5.04i)T - 89iT^{2} \)
97 \( 1 + (-3.40 + 0.676i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.65242930814822340519211145747, −11.65947046034222677974875021157, −10.59214123852953699195231069017, −9.135135615861197481575711443447, −7.992097020417714318768102415023, −6.75447307522417822299712418040, −6.33247696758781809295643734485, −4.80398237322794897332938755348, −3.46431933431725610926060225032, −0.14917322567801537058857354901, 3.04855672542254933508785539632, 4.22440030108754148933926475122, 5.43715169094322064354344198451, 6.60314197478936352184044096831, 8.092580236205174336773648196873, 9.620313275449744942935560030371, 10.24636480463054745811592221876, 11.45833712125196828814298571820, 11.80813935112079252792315313256, 12.90360356583739831194769150471

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