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Properties

Label 2-170-85.28-c1-0-1
Degree $2$
Conductor $170$
Sign $0.00921 - 0.999i$
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  + (−0.923 + 0.382i)2-s + (0.165 − 0.247i)3-s + (0.707 − 0.707i)4-s + (−2.07 + 0.831i)5-s + (−0.0581 + 0.292i)6-s + (−0.640 + 3.22i)7-s + (−0.382 + 0.923i)8-s + (1.11 + 2.68i)9-s + (1.59 − 1.56i)10-s + (2.53 + 0.504i)11-s + (−0.0581 − 0.292i)12-s + 1.91·13-s + (−0.640 − 3.22i)14-s + (−0.137 + 0.652i)15-s − i·16-s + (−2.84 + 2.98i)17-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (0.0956 − 0.143i)3-s + (0.353 − 0.353i)4-s + (−0.928 + 0.371i)5-s + (−0.0237 + 0.119i)6-s + (−0.242 + 1.21i)7-s + (−0.135 + 0.326i)8-s + (0.371 + 0.896i)9-s + (0.505 − 0.494i)10-s + (0.765 + 0.152i)11-s + (−0.0167 − 0.0844i)12-s + 0.531·13-s + (−0.171 − 0.860i)14-s + (−0.0355 + 0.168i)15-s − 0.250i·16-s + (−0.690 + 0.723i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00921 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00921 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.00921 - 0.999i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.00921 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.512635 + 0.507933i\)
\(L(\frac12)\) \(\approx\) \(0.512635 + 0.507933i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (2.07 - 0.831i)T \)
17 \( 1 + (2.84 - 2.98i)T \)
good3 \( 1 + (-0.165 + 0.247i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (0.640 - 3.22i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (-2.53 - 0.504i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 - 1.91T + 13T^{2} \)
19 \( 1 + (0.0675 - 0.163i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (3.34 - 2.23i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (7.00 + 4.68i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (-7.76 + 1.54i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (3.61 + 2.41i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-1.17 + 0.784i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-6.31 - 2.61i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 7.66iT - 47T^{2} \)
53 \( 1 + (-1.08 - 2.61i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-13.0 + 5.38i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.752 + 1.12i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (-5.06 - 5.06i)T + 67iT^{2} \)
71 \( 1 + (-2.15 - 10.8i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (1.89 + 9.52i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (2.20 - 11.1i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (-7.27 + 3.01i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (8.71 - 8.71i)T - 89iT^{2} \)
97 \( 1 + (1.09 + 5.49i)T + (-89.6 + 37.1i)T^{2} \)
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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

−12.87937921130940762081754242292, −11.81430080804793754048643567220, −11.10445886539619243171138874176, −9.916138995755393682051451755882, −8.737167303439681315291081213406, −8.023854240236390690002107342032, −6.88538656226867579103725636520, −5.78106671827947760258673795639, −4.04653778031690703957595072227, −2.21200648139040750445895749045, 0.874723968561616179413771393662, 3.52484778444534398432264438697, 4.32026611599447304438668100781, 6.56805660118326995460923468649, 7.36431995624296536586800198443, 8.592824436139069104345690063385, 9.424006969205590626550637118802, 10.53591473326152814533108157080, 11.46161459365303743825461942674, 12.30302347162113280204787721491

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