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Properties

Label 2-170-85.64-c1-0-4
Degree $2$
Conductor $170$
Sign $0.771 - 0.635i$
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  + 2-s + (−1 + i)3-s + 4-s + (2 + i)5-s + (−1 + i)6-s + (−1 − i)7-s + 8-s + i·9-s + (2 + i)10-s + (1 − i)11-s + (−1 + i)12-s + 4i·13-s + (−1 − i)14-s + (−3 + i)15-s + 16-s + (−1 − 4i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.577 + 0.577i)3-s + 0.5·4-s + (0.894 + 0.447i)5-s + (−0.408 + 0.408i)6-s + (−0.377 − 0.377i)7-s + 0.353·8-s + 0.333i·9-s + (0.632 + 0.316i)10-s + (0.301 − 0.301i)11-s + (−0.288 + 0.288i)12-s + 1.10i·13-s + (−0.267 − 0.267i)14-s + (−0.774 + 0.258i)15-s + 0.250·16-s + (−0.242 − 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.771 - 0.635i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.771 - 0.635i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46936 + 0.527401i\)
\(L(\frac12)\) \(\approx\) \(1.46936 + 0.527401i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (-2 - i)T \)
17 \( 1 + (1 + 4i)T \)
good3 \( 1 + (1 - i)T - 3iT^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + (5 + 5i)T + 29iT^{2} \)
31 \( 1 + (3 + 3i)T + 31iT^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 + (7 - 7i)T - 41iT^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 - 10iT - 59T^{2} \)
61 \( 1 + (-3 + 3i)T - 61iT^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + (-5 - 5i)T + 71iT^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + (1 - i)T - 79iT^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−13.27353481092918379353726537017, −11.59175992384993416989578335278, −11.12857760432409557302781387175, −10.01700834623811127043438610429, −9.185269514154034219715538313642, −7.23329333661608896703188277189, −6.34227775818918733068350532003, −5.24990602830898888969316293816, −4.14816207664015964405585001008, −2.44388586731091253164494743870, 1.71876909499902233373490341764, 3.62060212308278844401951532335, 5.46544291666398114091288094250, 5.95365692016010504537382071443, 7.05567054220422885335678517954, 8.547946120723644992216246125955, 9.809674527744071840781673180327, 10.78214039284368841685183049322, 12.24792255173595288032568015442, 12.53598140007763696511886402822

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