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Properties

Label 2-170-85.9-c1-0-3
Degree $2$
Conductor $170$
Sign $-0.339 - 0.940i$
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  + (0.707 + 0.707i)2-s + (−0.961 + 2.32i)3-s + 1.00i·4-s + (2.19 − 0.434i)5-s + (−2.32 + 0.961i)6-s + (0.124 − 0.0515i)7-s + (−0.707 + 0.707i)8-s + (−2.34 − 2.34i)9-s + (1.85 + 1.24i)10-s + (−1.51 + 0.627i)11-s + (−2.32 − 0.961i)12-s − 3.30·13-s + (0.124 + 0.0515i)14-s + (−1.09 + 5.50i)15-s − 1.00·16-s + (2.59 − 3.20i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.555 + 1.33i)3-s + 0.500i·4-s + (0.980 − 0.194i)5-s + (−0.947 + 0.392i)6-s + (0.0470 − 0.0194i)7-s + (−0.250 + 0.250i)8-s + (−0.780 − 0.780i)9-s + (0.587 + 0.393i)10-s + (−0.456 + 0.189i)11-s + (−0.669 − 0.277i)12-s − 0.917·13-s + (0.0332 + 0.0137i)14-s + (−0.283 + 1.42i)15-s − 0.250·16-s + (0.629 − 0.777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.339 - 0.940i$
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: \((2,\ 170,\ (\ :1/2),\ -0.339 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788356 + 1.12324i\)
\(L(\frac12)\) \(\approx\) \(0.788356 + 1.12324i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 + 0.434i)T \)
17 \( 1 + (-2.59 + 3.20i)T \)
good3 \( 1 + (0.961 - 2.32i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-0.124 + 0.0515i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.51 - 0.627i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 3.30T + 13T^{2} \)
19 \( 1 + (-3.24 + 3.24i)T - 19iT^{2} \)
23 \( 1 + (-1.86 - 4.49i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-2.75 + 6.65i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-6.99 - 2.89i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-1.21 + 2.92i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.66 - 4.02i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.56 - 1.56i)T - 43iT^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 + (5.73 + 5.73i)T + 53iT^{2} \)
59 \( 1 + (0.746 + 0.746i)T + 59iT^{2} \)
61 \( 1 + (1.57 + 3.80i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 11.9iT - 67T^{2} \)
71 \( 1 + (12.0 + 5.00i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-14.0 - 5.82i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-6.14 + 2.54i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (0.434 + 0.434i)T + 83iT^{2} \)
89 \( 1 - 12.1iT - 89T^{2} \)
97 \( 1 + (-0.683 - 0.283i)T + (68.5 + 68.5i)T^{2} \)
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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

−13.26366092590389932717755605010, −12.04819222009997444967118476477, −11.07087439680141197022210215474, −9.711721147119163356441393512587, −9.600875185748638588859339377518, −7.83371214159713584085901039926, −6.40663422592446468814092038829, −5.11918194194550632810229686579, −4.83733144710430643670524856189, −2.98692896544378499035718436380, 1.45013106543187212715768579139, 2.81778668887716344194010782236, 5.07109932423910200093257570082, 6.02958358058715625379186542959, 6.91218523216058304179768790914, 8.165253053536496184635768450168, 9.795523896559497029995849420348, 10.59856366674142771080540222449, 11.82222023986186807674884844081, 12.54880983172767724900044066533

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