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Properties

Label 2-170-85.57-c1-0-7
Degree $2$
Conductor $170$
Sign $0.908 + 0.417i$
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 + (0.942 − 1.41i)3-s + (−0.707 + 0.707i)4-s + (−0.577 − 2.16i)5-s + (1.66 + 0.330i)6-s + (0.999 − 5.02i)7-s + (−0.923 − 0.382i)8-s + (0.0471 + 0.113i)9-s + (1.77 − 1.35i)10-s + (−0.999 + 5.02i)11-s + (0.330 + 1.66i)12-s + 3.13i·13-s + (5.02 − 0.999i)14-s + (−3.59 − 1.22i)15-s − i·16-s + (2.02 + 3.59i)17-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (0.544 − 0.814i)3-s + (−0.353 + 0.353i)4-s + (−0.258 − 0.966i)5-s + (0.679 + 0.135i)6-s + (0.377 − 1.89i)7-s + (−0.326 − 0.135i)8-s + (0.0157 + 0.0379i)9-s + (0.561 − 0.430i)10-s + (−0.301 + 1.51i)11-s + (0.0955 + 0.480i)12-s + 0.870i·13-s + (1.34 − 0.267i)14-s + (−0.927 − 0.315i)15-s − 0.250i·16-s + (0.490 + 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.908 + 0.417i$
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: \((2,\ 170,\ (\ :1/2),\ 0.908 + 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41402 - 0.308965i\)
\(L(\frac12)\) \(\approx\) \(1.41402 - 0.308965i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.577 + 2.16i)T \)
17 \( 1 + (-2.02 - 3.59i)T \)
good3 \( 1 + (-0.942 + 1.41i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (-0.999 + 5.02i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (0.999 - 5.02i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 - 3.13iT - 13T^{2} \)
19 \( 1 + (-0.297 + 0.718i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.489 + 0.327i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-1.78 + 2.66i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (0.00940 + 0.0472i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (1.83 + 1.22i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-0.774 - 1.15i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-1.05 + 2.55i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 1.90T + 47T^{2} \)
53 \( 1 + (10.6 - 4.43i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.75 - 1.14i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (8.20 - 5.48i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (-7.07 + 7.07i)T - 67iT^{2} \)
71 \( 1 + (-7.55 + 1.50i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (1.39 + 7.03i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (9.46 + 1.88i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (-6.48 - 15.6i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (5.07 - 5.07i)T - 89iT^{2} \)
97 \( 1 + (-0.353 - 1.77i)T + (-89.6 + 37.1i)T^{2} \)
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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

−12.91893392484610694390333300214, −12.20571293430670384508913937768, −10.64935534905402482193802034369, −9.489563637377884196047664519412, −8.080360160676637586616138275563, −7.57724973808373031868915202360, −6.76578114919167894179643330366, −4.80166758666600098544423139267, −4.09482410980276287912280167701, −1.55522272741918426597009621351, 2.79986084220572845748567587414, 3.30369051271733020827166126532, 5.14009680568428114493043714976, 6.10655495540863887202357559019, 8.093335872682560050753129991742, 8.924725307308228393689608984184, 9.897134725343429961843860144342, 10.95754730755389309925738112762, 11.69218500089096269528207339257, 12.64989673552716649179170347747

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