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Properties

Label 2-170-17.15-c1-0-2
Degree $2$
Conductor $170$
Sign $0.885 + 0.465i$
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 + (−0.0294 − 0.0709i)3-s + 1.00i·4-s + (−0.923 + 0.382i)5-s + (−0.0294 + 0.0709i)6-s + (3.48 + 1.44i)7-s + (0.707 − 0.707i)8-s + (2.11 − 2.11i)9-s + (0.923 + 0.382i)10-s + (1.39 − 3.37i)11-s + (0.0709 − 0.0294i)12-s + 4.29i·13-s + (−1.44 − 3.48i)14-s + (0.0543 + 0.0543i)15-s − 1.00·16-s + (2.96 − 2.86i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.0169 − 0.0409i)3-s + 0.500i·4-s + (−0.413 + 0.171i)5-s + (−0.0120 + 0.0289i)6-s + (1.31 + 0.546i)7-s + (0.250 − 0.250i)8-s + (0.705 − 0.705i)9-s + (0.292 + 0.121i)10-s + (0.421 − 1.01i)11-s + (0.0204 − 0.00848i)12-s + 1.19i·13-s + (−0.386 − 0.932i)14-s + (0.0140 + 0.0140i)15-s − 0.250·16-s + (0.718 − 0.696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.885 + 0.465i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.885 + 0.465i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.962118 - 0.237592i\)
\(L(\frac12)\) \(\approx\) \(0.962118 - 0.237592i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 + (-2.96 + 2.86i)T \)
good3 \( 1 + (0.0294 + 0.0709i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (-3.48 - 1.44i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.39 + 3.37i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 4.29iT - 13T^{2} \)
19 \( 1 + (2.18 + 2.18i)T + 19iT^{2} \)
23 \( 1 + (2.47 - 5.98i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-2.10 + 0.871i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-2.31 - 5.58i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (3.10 + 7.49i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (7.61 + 3.15i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (7.80 - 7.80i)T - 43iT^{2} \)
47 \( 1 - 8.27iT - 47T^{2} \)
53 \( 1 + (7.36 + 7.36i)T + 53iT^{2} \)
59 \( 1 + (3.39 - 3.39i)T - 59iT^{2} \)
61 \( 1 + (2.56 + 1.06i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 3.43T + 67T^{2} \)
71 \( 1 + (5.25 + 12.6i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-10.9 + 4.53i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.15 + 2.80i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (1.30 + 1.30i)T + 83iT^{2} \)
89 \( 1 - 9.64iT - 89T^{2} \)
97 \( 1 + (0.613 - 0.254i)T + (68.5 - 68.5i)T^{2} \)
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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

−12.18869462853600879718178528536, −11.68755064872605309689219219163, −10.95262382147450411212884691876, −9.557770404544890043202703891275, −8.725725171995849662238897941454, −7.72775018042051086988705359343, −6.52943823059306203441378910511, −4.82610255980682381290886194147, −3.48789258375340691781226307507, −1.57124505758041150929784473835, 1.62336883238961192903559128684, 4.24336063277552342083638138420, 5.15303798970325904065632917820, 6.80368706150279002961889988224, 7.996518555374194391316048728988, 8.227118791302506665496513562051, 10.14845623117433491917440763084, 10.45023426969981828487711011347, 11.81350673300795919821650299180, 12.81444158509702673141441832751

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