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Properties

Label 2-170-85.57-c1-0-8
Degree $2$
Conductor $170$
Sign $-0.649 + 0.760i$
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 − 2.33i)3-s + (−0.707 + 0.707i)4-s + (−2.19 + 0.434i)5-s + (−2.75 − 0.548i)6-s + (0.254 − 1.27i)7-s + (0.923 + 0.382i)8-s + (−1.88 − 4.54i)9-s + (1.24 + 1.86i)10-s + (1.14 − 5.73i)11-s + (0.548 + 2.75i)12-s + 5.33i·13-s + (−1.27 + 0.254i)14-s + (−2.41 + 5.80i)15-s − i·16-s + (1.22 + 3.93i)17-s + ⋯
L(s)  = 1  + (−0.270 − 0.653i)2-s + (0.902 − 1.35i)3-s + (−0.353 + 0.353i)4-s + (−0.980 + 0.194i)5-s + (−1.12 − 0.224i)6-s + (0.0961 − 0.483i)7-s + (0.326 + 0.135i)8-s + (−0.626 − 1.51i)9-s + (0.392 + 0.588i)10-s + (0.343 − 1.72i)11-s + (0.158 + 0.796i)12-s + 1.48i·13-s + (−0.341 + 0.0679i)14-s + (−0.622 + 1.50i)15-s − 0.250i·16-s + (0.296 + 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.649 + 0.760i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.463674 - 1.00537i\)
\(L(\frac12)\) \(\approx\) \(0.463674 - 1.00537i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 - 0.434i)T \)
17 \( 1 + (-1.22 - 3.93i)T \)
good3 \( 1 + (-1.56 + 2.33i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (-0.254 + 1.27i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (-1.14 + 5.73i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 - 5.33iT - 13T^{2} \)
19 \( 1 + (0.311 - 0.752i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-5.82 + 3.88i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (1.17 - 1.75i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-0.301 - 1.51i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-5.79 - 3.87i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (4.40 + 6.59i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.544 - 1.31i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 0.109T + 47T^{2} \)
53 \( 1 + (-0.915 + 0.379i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (5.08 - 2.10i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (4.82 - 3.22i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (4.12 - 4.12i)T - 67iT^{2} \)
71 \( 1 + (5.38 - 1.07i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-1.59 - 8.02i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (5.01 + 0.997i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (-0.987 - 2.38i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-0.779 + 0.779i)T - 89iT^{2} \)
97 \( 1 + (2.93 + 14.7i)T + (-89.6 + 37.1i)T^{2} \)
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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

−12.37121264362362677211387847339, −11.53875298009630130630280966979, −10.72150389608410356394874252965, −8.871085626892048101851936826360, −8.466830529914606357908669669719, −7.39568251714396265344203174162, −6.47110131845952476620478757760, −4.02965358391726745604689214905, −2.96912531461820396867752387500, −1.20252244561662292495573947137, 3.06717878142844332821078812920, 4.43085928288959511562986837719, 5.21756572869671188716234270067, 7.30173480218098648922833864558, 8.053939022668008849830044098216, 9.172122150827703636986938748096, 9.731915663078163870781660299480, 10.85095675620452550177180198772, 12.13574255905278752499871713836, 13.29436845347719427834165333102

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