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Properties

Label 2-170-17.13-c1-0-1
Degree $2$
Conductor $170$
Sign $-0.337 - 0.941i$
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  + i·2-s + (1.90 + 1.90i)3-s − 4-s + (−0.707 − 0.707i)5-s + (−1.90 + 1.90i)6-s + (−2.69 + 2.69i)7-s − i·8-s + 4.28i·9-s + (0.707 − 0.707i)10-s + (3.82 − 3.82i)11-s + (−1.90 − 1.90i)12-s + 2.11·13-s + (−2.69 − 2.69i)14-s − 2.69i·15-s + 16-s + (0.908 + 4.02i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (1.10 + 1.10i)3-s − 0.5·4-s + (−0.316 − 0.316i)5-s + (−0.779 + 0.779i)6-s + (−1.02 + 1.02i)7-s − 0.353i·8-s + 1.42i·9-s + (0.223 − 0.223i)10-s + (1.15 − 1.15i)11-s + (−0.550 − 0.550i)12-s + 0.586·13-s + (−0.721 − 0.721i)14-s − 0.696i·15-s + 0.250·16-s + (0.220 + 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.337 - 0.941i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.337 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794565 + 1.12945i\)
\(L(\frac12)\) \(\approx\) \(0.794565 + 1.12945i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (-0.908 - 4.02i)T \)
good3 \( 1 + (-1.90 - 1.90i)T + 3iT^{2} \)
7 \( 1 + (2.69 - 2.69i)T - 7iT^{2} \)
11 \( 1 + (-3.82 + 3.82i)T - 11iT^{2} \)
13 \( 1 - 2.11T + 13T^{2} \)
19 \( 1 + 7.10iT - 19T^{2} \)
23 \( 1 + (0.585 - 0.585i)T - 23iT^{2} \)
29 \( 1 + (-3.49 - 3.49i)T + 29iT^{2} \)
31 \( 1 + (0.779 + 0.779i)T + 31iT^{2} \)
37 \( 1 + (7.52 + 7.52i)T + 37iT^{2} \)
41 \( 1 + (4.39 - 4.39i)T - 41iT^{2} \)
43 \( 1 + 7.81iT - 43T^{2} \)
47 \( 1 - 2.29T + 47T^{2} \)
53 \( 1 - 3.28iT - 53T^{2} \)
59 \( 1 + 0.555iT - 59T^{2} \)
61 \( 1 + (-2.89 + 2.89i)T - 61iT^{2} \)
67 \( 1 + 3.97T + 67T^{2} \)
71 \( 1 + (4.59 + 4.59i)T + 71iT^{2} \)
73 \( 1 + (-7.45 - 7.45i)T + 73iT^{2} \)
79 \( 1 + (9.61 - 9.61i)T - 79iT^{2} \)
83 \( 1 + 4.04iT - 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (0.600 + 0.600i)T + 97iT^{2} \)
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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

−13.40126515331847543877710482935, −12.25321447287575569977713250755, −10.87934673040161525370862991331, −9.564498281594964215173031832887, −8.754567250965098015484978996225, −8.586226745510843205797088235719, −6.71078285704300113752764645137, −5.55043117212567397661698147312, −4.01089029741409388096530864427, −3.15259502237055864102452853821, 1.48595873524554123715916416287, 3.13924273213518794600412410638, 4.04862303143698176813781001439, 6.54961469742796281192202104913, 7.26882916472911015860829651482, 8.340997996793908579974026183061, 9.554203083468270187585305185182, 10.26509450651480193370779287239, 11.87445792111913603056414294158, 12.45250052617633257163546823329

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