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Properties

Label 2-170-85.49-c1-0-8
Degree $2$
Conductor $170$
Sign $-0.388 + 0.921i$
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 + (−1.99 + 0.826i)3-s − 1.00i·4-s + (−1.27 − 1.83i)5-s + (−0.826 + 1.99i)6-s + (1.32 − 3.20i)7-s + (−0.707 − 0.707i)8-s + (1.18 − 1.18i)9-s + (−2.20 − 0.399i)10-s + (1.63 − 3.94i)11-s + (0.826 + 1.99i)12-s − 5.21·13-s + (−1.32 − 3.20i)14-s + (4.06 + 2.61i)15-s − 1.00·16-s + (2.46 + 3.30i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−1.15 + 0.477i)3-s − 0.500i·4-s + (−0.569 − 0.821i)5-s + (−0.337 + 0.814i)6-s + (0.501 − 1.21i)7-s + (−0.250 − 0.250i)8-s + (0.393 − 0.393i)9-s + (−0.695 − 0.126i)10-s + (0.492 − 1.18i)11-s + (0.238 + 0.576i)12-s − 1.44·13-s + (−0.354 − 0.856i)14-s + (1.04 + 0.675i)15-s − 0.250·16-s + (0.598 + 0.800i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.388 + 0.921i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479403 - 0.722215i\)
\(L(\frac12)\) \(\approx\) \(0.479403 - 0.722215i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.27 + 1.83i)T \)
17 \( 1 + (-2.46 - 3.30i)T \)
good3 \( 1 + (1.99 - 0.826i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (-1.32 + 3.20i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.63 + 3.94i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 5.21T + 13T^{2} \)
19 \( 1 + (0.305 + 0.305i)T + 19iT^{2} \)
23 \( 1 + (-6.86 - 2.84i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.46 + 1.43i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.648 + 1.56i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (4.56 - 1.89i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (4.20 + 1.74i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.38 - 2.38i)T + 43iT^{2} \)
47 \( 1 - 3.47T + 47T^{2} \)
53 \( 1 + (-9.73 + 9.73i)T - 53iT^{2} \)
59 \( 1 + (-6.37 + 6.37i)T - 59iT^{2} \)
61 \( 1 + (-5.75 - 2.38i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 0.409iT - 67T^{2} \)
71 \( 1 + (3.84 + 9.28i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-0.401 - 0.970i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-0.218 + 0.527i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (10.6 - 10.6i)T - 83iT^{2} \)
89 \( 1 - 14.5iT - 89T^{2} \)
97 \( 1 + (3.70 + 8.95i)T + (-68.5 + 68.5i)T^{2} \)
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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

−12.18953634477599826858187812157, −11.48376373000845791046203360713, −10.78407563286424769181170540057, −9.856188886152973221923766232499, −8.392584429564341647549251223445, −7.05014385590434116580946224037, −5.51693868565675624826799290393, −4.75337249044841968959498289699, −3.72245223454228299051968205988, −0.831175054418013770911424493248, 2.64555865325185503041679445199, 4.71331481564894180826155702787, 5.53783909693141869157948314328, 6.91489961402879744358625119261, 7.28615873764959480818884828069, 8.876798132053265728950210024697, 10.32581923493203918109750766077, 11.67602879820073948333560875617, 12.01149946103831767827088302222, 12.64050734486148616947902202083

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