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Properties

Label 2-170-17.4-c1-0-1
Degree $2$
Conductor $170$
Sign $0.151 - 0.988i$
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  + i·2-s + (0.292 − 0.292i)3-s − 4-s + (−0.707 + 0.707i)5-s + (0.292 + 0.292i)6-s + (2.41 + 2.41i)7-s − i·8-s + 2.82i·9-s + (−0.707 − 0.707i)10-s + (1 + i)11-s + (−0.292 + 0.292i)12-s + 13-s + (−2.41 + 2.41i)14-s + 0.414i·15-s + 16-s + (0.121 − 4.12i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.169 − 0.169i)3-s − 0.5·4-s + (−0.316 + 0.316i)5-s + (0.119 + 0.119i)6-s + (0.912 + 0.912i)7-s − 0.353i·8-s + 0.942i·9-s + (−0.223 − 0.223i)10-s + (0.301 + 0.301i)11-s + (−0.0845 + 0.0845i)12-s + 0.277·13-s + (−0.645 + 0.645i)14-s + 0.106i·15-s + 0.250·16-s + (0.0294 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.151 - 0.988i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.151 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.891756 + 0.765620i\)
\(L(\frac12)\) \(\approx\) \(0.891756 + 0.765620i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-0.121 + 4.12i)T \)
good3 \( 1 + (-0.292 + 0.292i)T - 3iT^{2} \)
7 \( 1 + (-2.41 - 2.41i)T + 7iT^{2} \)
11 \( 1 + (-1 - i)T + 11iT^{2} \)
13 \( 1 - T + 13T^{2} \)
19 \( 1 + 0.414iT - 19T^{2} \)
23 \( 1 + (6.24 + 6.24i)T + 23iT^{2} \)
29 \( 1 + (-2.94 + 2.94i)T - 29iT^{2} \)
31 \( 1 + (2.29 - 2.29i)T - 31iT^{2} \)
37 \( 1 + (-4.41 + 4.41i)T - 37iT^{2} \)
41 \( 1 + (-4.65 - 4.65i)T + 41iT^{2} \)
43 \( 1 + 1.75iT - 43T^{2} \)
47 \( 1 + 5.24T + 47T^{2} \)
53 \( 1 + 13.4iT - 53T^{2} \)
59 \( 1 + 8.89iT - 59T^{2} \)
61 \( 1 + (-9.77 - 9.77i)T + 61iT^{2} \)
67 \( 1 - 3.17T + 67T^{2} \)
71 \( 1 + (9.70 - 9.70i)T - 71iT^{2} \)
73 \( 1 + (1.53 - 1.53i)T - 73iT^{2} \)
79 \( 1 + (0.242 + 0.242i)T + 79iT^{2} \)
83 \( 1 - 9.89iT - 83T^{2} \)
89 \( 1 - 3.34T + 89T^{2} \)
97 \( 1 + (-5.87 + 5.87i)T - 97iT^{2} \)
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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

−13.13667101682083599415049482354, −11.99307353029809703661120530550, −11.13617228257861147013775922223, −9.848442544984779108762381453762, −8.525270768809875755844243949065, −7.956961461346961558441536091002, −6.79240118120070287131490287461, −5.46926401838269434235623132259, −4.40992379291363884799189249200, −2.35752891326219131099791633826, 1.33638061679884617982481031307, 3.59832864611256911667741440614, 4.37000080387952510118033450531, 5.99452000047757333420510181122, 7.63041495990691828633868422873, 8.552327072171080901715033195814, 9.626848304924193072666871902559, 10.67625062805330281044876081553, 11.54421682894597335275338641806, 12.36267076308963755038607728671

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