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Properties

Label 2-170-85.59-c1-0-9
Degree $2$
Conductor $170$
Sign $0.0199 + 0.999i$
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  + (0.707 + 0.707i)2-s + (−2.33 − 0.969i)3-s + 1.00i·4-s + (−0.309 − 2.21i)5-s + (−0.969 − 2.33i)6-s + (−1.26 − 3.06i)7-s + (−0.707 + 0.707i)8-s + (2.41 + 2.41i)9-s + (1.34 − 1.78i)10-s + (−0.949 − 2.29i)11-s + (0.969 − 2.33i)12-s − 0.777·13-s + (1.26 − 3.06i)14-s + (−1.42 + 5.48i)15-s − 1.00·16-s + (4.04 + 0.798i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−1.35 − 0.559i)3-s + 0.500i·4-s + (−0.138 − 0.990i)5-s + (−0.395 − 0.955i)6-s + (−0.479 − 1.15i)7-s + (−0.250 + 0.250i)8-s + (0.804 + 0.804i)9-s + (0.425 − 0.564i)10-s + (−0.286 − 0.691i)11-s + (0.279 − 0.675i)12-s − 0.215·13-s + (0.338 − 0.817i)14-s + (−0.367 + 1.41i)15-s − 0.250·16-s + (0.981 + 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0199 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0199 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.0199 + 0.999i$
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: \((2,\ 170,\ (\ :1/2),\ 0.0199 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540502 - 0.529805i\)
\(L(\frac12)\) \(\approx\) \(0.540502 - 0.529805i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 2.21i)T \)
17 \( 1 + (-4.04 - 0.798i)T \)
good3 \( 1 + (2.33 + 0.969i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (1.26 + 3.06i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.949 + 2.29i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 0.777T + 13T^{2} \)
19 \( 1 + (-4.13 + 4.13i)T - 19iT^{2} \)
23 \( 1 + (7.45 - 3.08i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.319 - 0.132i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (0.719 - 1.73i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-4.26 - 1.76i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.17 + 0.900i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-8.03 + 8.03i)T - 43iT^{2} \)
47 \( 1 - 9.25T + 47T^{2} \)
53 \( 1 + (2.98 + 2.98i)T + 53iT^{2} \)
59 \( 1 + (9.06 + 9.06i)T + 59iT^{2} \)
61 \( 1 + (-12.7 + 5.29i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 0.862iT - 67T^{2} \)
71 \( 1 + (3.13 - 7.57i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-3.75 + 9.05i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-3.33 - 8.04i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (2.56 + 2.56i)T + 83iT^{2} \)
89 \( 1 - 3.67iT - 89T^{2} \)
97 \( 1 + (1.05 - 2.53i)T + (-68.5 - 68.5i)T^{2} \)
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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

−12.50806069414869030226461842384, −11.84478650748217500743285172484, −10.81998776033710333506026268256, −9.575625364746501395512297504926, −7.959747218004352363358519432804, −7.14120225574390858014770984678, −5.93031747900840457719932128559, −5.18420868327507210468483554160, −3.83218629172731629671496636526, −0.71853114342535793788387302193, 2.66093855540065972988396545384, 4.19718849389193770932782808534, 5.66295698564973685667909285553, 6.06500440239595487458338263628, 7.60708286395091444191548908370, 9.661035308713815331141738060409, 10.15791257780175086033213462240, 11.16260153966617651221535510018, 12.15715434755852942493657882165, 12.33066440020172525109278277957

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