Properties

Label 2-170-85.12-c1-0-2
Degree $2$
Conductor $170$
Sign $0.443 - 0.896i$
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 + (0.182 − 0.917i)3-s + (−0.707 − 0.707i)4-s + (−0.566 + 2.16i)5-s + (0.778 + 0.519i)6-s + (1.62 + 1.08i)7-s + (0.923 − 0.382i)8-s + (1.96 + 0.812i)9-s + (−1.78 − 1.35i)10-s + (−2.02 + 3.02i)11-s + (−0.778 + 0.519i)12-s + 6.15·13-s + (−1.62 + 1.08i)14-s + (1.88 + 0.915i)15-s + i·16-s + (−0.742 − 4.05i)17-s + ⋯
L(s)  = 1  + (−0.270 + 0.653i)2-s + (0.105 − 0.529i)3-s + (−0.353 − 0.353i)4-s + (−0.253 + 0.967i)5-s + (0.317 + 0.212i)6-s + (0.614 + 0.410i)7-s + (0.326 − 0.135i)8-s + (0.654 + 0.270i)9-s + (−0.563 − 0.427i)10-s + (−0.609 + 0.911i)11-s + (−0.224 + 0.150i)12-s + 1.70·13-s + (−0.434 + 0.290i)14-s + (0.485 + 0.236i)15-s + 0.250i·16-s + (−0.180 − 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.443 - 0.896i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.443 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.896725 + 0.556960i\)
\(L(\frac12)\) \(\approx\) \(0.896725 + 0.556960i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.566 - 2.16i)T \)
17 \( 1 + (0.742 + 4.05i)T \)
good3 \( 1 + (-0.182 + 0.917i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (-1.62 - 1.08i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (2.02 - 3.02i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 - 6.15T + 13T^{2} \)
19 \( 1 + (2.19 - 0.909i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (3.52 - 0.700i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (5.01 + 0.996i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (0.311 + 0.466i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (-4.49 - 0.893i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-4.93 + 0.981i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (3.01 + 7.27i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 12.6iT - 47T^{2} \)
53 \( 1 + (-6.86 - 2.84i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (5.12 - 12.3i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.856 - 4.30i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-5.21 + 5.21i)T - 67iT^{2} \)
71 \( 1 + (-4.65 + 3.11i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (1.98 - 1.32i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (11.1 + 7.44i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (4.02 - 9.70i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-1.02 - 1.02i)T + 89iT^{2} \)
97 \( 1 + (13.1 - 8.76i)T + (37.1 - 89.6i)T^{2} \)
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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

−13.23661817067739869243518096850, −11.91365389865048252846534040176, −10.87201008568915853077976675675, −9.966319028724921606580890313373, −8.551526980712028267602715191924, −7.63223496039183656851476063721, −6.89435658113380494579233094723, −5.69019736742569468842208275384, −4.13243027576021186741908753441, −2.07794411326717826578391494957, 1.32378200971959358858570210426, 3.69143760788615285511395949375, 4.47298084337615316221726636170, 5.98689768281907775194092086966, 7.956025526137160666721051468844, 8.555711541316931160639332450038, 9.582003914527649493270797554911, 10.83833059195743116088800883132, 11.22766395785023270580625410893, 12.78260064807445608593220147824

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