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Properties

Label 2-170-5.4-c1-0-7
Degree $2$
Conductor $170$
Sign $-0.981 + 0.193i$
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 − 2.62i·3-s − 4-s + (−2.19 + 0.432i)5-s − 2.62·6-s − 0.864i·7-s + i·8-s − 3.89·9-s + (0.432 + 2.19i)10-s + 2·11-s + 2.62i·12-s − 2.62i·13-s − 0.864·14-s + (1.13 + 5.76i)15-s + 16-s − i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.51i·3-s − 0.5·4-s + (−0.981 + 0.193i)5-s − 1.07·6-s − 0.326i·7-s + 0.353i·8-s − 1.29·9-s + (0.136 + 0.693i)10-s + 0.603·11-s + 0.758i·12-s − 0.728i·13-s − 0.231·14-s + (0.293 + 1.48i)15-s + 0.250·16-s − 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.981 + 0.193i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.981 + 0.193i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0847773 - 0.868704i\)
\(L(\frac12)\) \(\approx\) \(0.0847773 - 0.868704i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 - 0.432i)T \)
17 \( 1 + iT \)
good3 \( 1 + 2.62iT - 3T^{2} \)
7 \( 1 + 0.864iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 2.62iT - 13T^{2} \)
19 \( 1 - 0.896T + 19T^{2} \)
23 \( 1 + 3.13iT - 23T^{2} \)
29 \( 1 + 9.49T + 29T^{2} \)
31 \( 1 - 9.01T + 31T^{2} \)
37 \( 1 + 10.1iT - 37T^{2} \)
41 \( 1 - 9.52T + 41T^{2} \)
43 \( 1 - 7.25iT - 43T^{2} \)
47 \( 1 - 10.4iT - 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 - 4.14T + 59T^{2} \)
61 \( 1 - 3.28T + 61T^{2} \)
67 \( 1 + 1.25iT - 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + 2.62iT - 73T^{2} \)
79 \( 1 - 7.91T + 79T^{2} \)
83 \( 1 + 4.20iT - 83T^{2} \)
89 \( 1 + 4.14T + 89T^{2} \)
97 \( 1 - 6.14iT - 97T^{2} \)
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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

−12.36842205951907214069956092773, −11.52545347494407516721509544921, −10.70391293688259130236409916422, −9.155709425324687895666645601237, −7.917033739959190743089543608626, −7.34055186917600676450419699443, −6.04710854354530212228159614333, −4.19432427576104334395428306499, −2.71559532928683093406419897610, −0.884507012432809817552582319346, 3.61438390408960659081220386830, 4.40192245312682557445476497338, 5.50271502319871732694512477673, 6.98670727984606341928992060994, 8.336683430195542352609619991497, 9.136811082596766290575326790315, 9.999763738310155034417005828203, 11.28246719490995972292289253355, 11.98126812167622559749761974332, 13.46805951194021540458772476324

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