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Properties

Label 2-170-85.59-c1-0-8
Degree $2$
Conductor $170$
Sign $-0.947 + 0.319i$
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 + (−0.613 − 0.254i)3-s + 1.00i·4-s + (−1.28 + 1.82i)5-s + (0.254 + 0.613i)6-s + (−1.97 − 4.76i)7-s + (0.707 − 0.707i)8-s + (−1.80 − 1.80i)9-s + (2.20 − 0.384i)10-s + (−0.737 − 1.78i)11-s + (0.254 − 0.613i)12-s − 3.78·13-s + (−1.97 + 4.76i)14-s + (1.25 − 0.795i)15-s − 1.00·16-s + (3.13 + 2.68i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.354 − 0.146i)3-s + 0.500i·4-s + (−0.575 + 0.818i)5-s + (0.103 + 0.250i)6-s + (−0.745 − 1.80i)7-s + (0.250 − 0.250i)8-s + (−0.603 − 0.603i)9-s + (0.696 − 0.121i)10-s + (−0.222 − 0.537i)11-s + (0.0733 − 0.177i)12-s − 1.04·13-s + (−0.527 + 1.27i)14-s + (0.323 − 0.205i)15-s − 0.250·16-s + (0.759 + 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.947 + 0.319i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.947 + 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0558989 - 0.340603i\)
\(L(\frac12)\) \(\approx\) \(0.0558989 - 0.340603i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 - 1.82i)T \)
17 \( 1 + (-3.13 - 2.68i)T \)
good3 \( 1 + (0.613 + 0.254i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (1.97 + 4.76i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.737 + 1.78i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 3.78T + 13T^{2} \)
19 \( 1 + (2.29 - 2.29i)T - 19iT^{2} \)
23 \( 1 + (-3.23 + 1.34i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (3.27 + 1.35i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.81 + 6.79i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (0.515 + 0.213i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.83 + 1.58i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (3.43 - 3.43i)T - 43iT^{2} \)
47 \( 1 - 4.24T + 47T^{2} \)
53 \( 1 + (3.39 + 3.39i)T + 53iT^{2} \)
59 \( 1 + (-6.83 - 6.83i)T + 59iT^{2} \)
61 \( 1 + (-1.47 + 0.610i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 + (-0.0895 + 0.216i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-4.37 + 10.5i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (3.43 + 8.28i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (7.42 + 7.42i)T + 83iT^{2} \)
89 \( 1 - 7.56iT - 89T^{2} \)
97 \( 1 + (3.16 - 7.64i)T + (-68.5 - 68.5i)T^{2} \)
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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

−12.16234704354180761963446308739, −11.12332382116101161575169602572, −10.46350039871881257036637303123, −9.646496997199140956365572882727, −8.022047622786064506032067617122, −7.20661001989698369486014476259, −6.21693866016557573335126344799, −4.05774530164954069314845384164, −3.10493501197939871635327500287, −0.37562004088688509317274810786, 2.62367705576302431229746128965, 5.02010250068206010935232171708, 5.50845751501380557181423366355, 7.03085231088543169984588500229, 8.274968753142134111769867937410, 9.067706582720004101795836548321, 9.897522864204136302901446248022, 11.38624676935186939991445401197, 12.20134431956088882534029149051, 12.90660916618063185236374032677

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