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Properties

Label 2-170-85.59-c1-0-4
Degree $2$
Conductor $170$
Sign $0.721 - 0.692i$
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 + (0.384 − 2.20i)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 + (1.82 − 1.28i)10-s + (−0.737 − 1.78i)11-s + (−0.254 + 0.613i)12-s + 3.78·13-s + (−1.97 + 4.76i)14-s + (0.795 − 1.25i)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 + (0.171 − 0.985i)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 + (0.578 − 0.406i)10-s + (−0.222 − 0.537i)11-s + (−0.0733 + 0.177i)12-s + 1.04·13-s + (−0.527 + 1.27i)14-s + (0.205 − 0.323i)15-s − 0.250·16-s + (−0.759 − 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.721 - 0.692i$
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: \((2,\ 170,\ (\ :1/2),\ 0.721 - 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52893 + 0.615332i\)
\(L(\frac12)\) \(\approx\) \(1.52893 + 0.615332i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.384 + 2.20i)T \)
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

−12.99248366368145646837419341196, −11.93403965539967015095264639028, −11.39421516555178505853474429720, −9.358954322792501620692503906906, −8.646714506172043302566025466361, −8.145284728825783289652166067040, −5.99431855217929021127630771337, −5.58842003252072344310859373267, −4.12941540911725896943749782753, −2.40474146391366377332109031772, 1.94777930836262254470726597481, 3.54285918101650405489592143448, 4.64880004954408670225523994849, 6.34021074390564470584458766475, 7.37685260516359548820266833235, 8.429026616189990746402139782602, 10.11913865554118367357877994896, 10.90332498169999355899687326911, 11.21087545086296328976964104158, 12.99144065041161289059488658255

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