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Properties

Label 2-170-85.9-c1-0-4
Degree $2$
Conductor $170$
Sign $0.502 - 0.864i$
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 + (0.0978 − 0.236i)3-s + 1.00i·4-s + (−1.19 + 1.89i)5-s + (0.236 − 0.0978i)6-s + (3.48 − 1.44i)7-s + (−0.707 + 0.707i)8-s + (2.07 + 2.07i)9-s + (−2.18 + 0.496i)10-s + (−3.09 + 1.28i)11-s + (0.236 + 0.0978i)12-s + 0.0184·13-s + (3.48 + 1.44i)14-s + (0.330 + 0.466i)15-s − 1.00·16-s + (−2.88 − 2.95i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.0565 − 0.136i)3-s + 0.500i·4-s + (−0.532 + 0.846i)5-s + (0.0964 − 0.0399i)6-s + (1.31 − 0.545i)7-s + (−0.250 + 0.250i)8-s + (0.691 + 0.691i)9-s + (−0.689 + 0.157i)10-s + (−0.933 + 0.386i)11-s + (0.0682 + 0.0282i)12-s + 0.00510·13-s + (0.931 + 0.385i)14-s + (0.0854 + 0.120i)15-s − 0.250·16-s + (−0.698 − 0.715i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.502 - 0.864i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29630 + 0.745500i\)
\(L(\frac12)\) \(\approx\) \(1.29630 + 0.745500i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 - 1.89i)T \)
17 \( 1 + (2.88 + 2.95i)T \)
good3 \( 1 + (-0.0978 + 0.236i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-3.48 + 1.44i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (3.09 - 1.28i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 0.0184T + 13T^{2} \)
19 \( 1 + (-4.04 + 4.04i)T - 19iT^{2} \)
23 \( 1 + (2.00 + 4.83i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.238 - 0.575i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.958 + 0.397i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.49 - 3.60i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.68 + 6.47i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (3.22 - 3.22i)T - 43iT^{2} \)
47 \( 1 - 9.78T + 47T^{2} \)
53 \( 1 + (9.03 + 9.03i)T + 53iT^{2} \)
59 \( 1 + (-10.1 - 10.1i)T + 59iT^{2} \)
61 \( 1 + (-2.19 - 5.30i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 + (3.61 + 1.49i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (9.33 + 3.86i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-4.92 + 2.03i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (0.848 + 0.848i)T + 83iT^{2} \)
89 \( 1 - 14.4iT - 89T^{2} \)
97 \( 1 + (6.49 + 2.69i)T + (68.5 + 68.5i)T^{2} \)
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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

−13.24702517345256779097196852683, −11.89770220604185333404467959369, −11.03580146718061748074662399431, −10.23189831895240847225329373401, −8.404204469252851281718409883936, −7.46953219227752691160623108264, −7.00424827744232494976905000736, −5.11841867708858133585037880244, −4.31540202745759144154707887152, −2.49460836697448024965527775952, 1.62664359164358536373950213352, 3.68108867302760834169604994613, 4.81633806019620614994139815929, 5.73656613380820650821145796527, 7.63471513851365660126742448396, 8.519170137495840962517757117189, 9.638800431344160577284619994644, 10.88378982017126458083798339571, 11.76951362874851934094990877354, 12.47045308044356163474626359248

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