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Properties

Label 2-170-17.2-c1-0-2
Degree $2$
Conductor $170$
Sign $0.651 + 0.758i$
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 + (−2.93 + 1.21i)3-s + 1.00i·4-s + (−0.382 − 0.923i)5-s + (2.93 + 1.21i)6-s + (0.165 − 0.400i)7-s + (0.707 − 0.707i)8-s + (5.02 − 5.02i)9-s + (−0.382 + 0.923i)10-s + (4.66 + 1.93i)11-s + (−1.21 − 2.93i)12-s − 5.61i·13-s + (−0.400 + 0.165i)14-s + (2.24 + 2.24i)15-s − 1.00·16-s + (3.76 + 1.68i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−1.69 + 0.702i)3-s + 0.500i·4-s + (−0.171 − 0.413i)5-s + (1.19 + 0.496i)6-s + (0.0627 − 0.151i)7-s + (0.250 − 0.250i)8-s + (1.67 − 1.67i)9-s + (−0.121 + 0.292i)10-s + (1.40 + 0.582i)11-s + (−0.351 − 0.847i)12-s − 1.55i·13-s + (−0.107 + 0.0443i)14-s + (0.580 + 0.580i)15-s − 0.250·16-s + (0.912 + 0.409i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.651 + 0.758i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.651 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.483379 - 0.222085i\)
\(L(\frac12)\) \(\approx\) \(0.483379 - 0.222085i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.382 + 0.923i)T \)
17 \( 1 + (-3.76 - 1.68i)T \)
good3 \( 1 + (2.93 - 1.21i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (-0.165 + 0.400i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-4.66 - 1.93i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 5.61iT - 13T^{2} \)
19 \( 1 + (4.34 + 4.34i)T + 19iT^{2} \)
23 \( 1 + (-4.74 - 1.96i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.02 + 2.46i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.02 + 0.840i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.12 + 0.464i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.90 + 4.60i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-0.638 + 0.638i)T - 43iT^{2} \)
47 \( 1 + 5.95iT - 47T^{2} \)
53 \( 1 + (-0.0241 - 0.0241i)T + 53iT^{2} \)
59 \( 1 + (-0.275 + 0.275i)T - 59iT^{2} \)
61 \( 1 + (-2.72 + 6.57i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 3.27T + 67T^{2} \)
71 \( 1 + (3.63 - 1.50i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.30 + 3.13i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (12.2 + 5.05i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.560 - 0.560i)T + 83iT^{2} \)
89 \( 1 - 2.73iT - 89T^{2} \)
97 \( 1 + (-4.95 - 11.9i)T + (-68.5 + 68.5i)T^{2} \)
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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

−12.31349181491477926071504647627, −11.52649946797756166111730905099, −10.68712870202953640870654394321, −9.933581276085125423967806827152, −8.923649986113461668211603557130, −7.33110261266940782406439881339, −6.09719184997104970543894877187, −4.91081263932246055338924645330, −3.81985245335990177083633183938, −0.870286003638498403098741954879, 1.37545382171209595839243013424, 4.40178890425310284812922761421, 5.86802303084355538261124184029, 6.55727349472595655071948175644, 7.28846127974340318602711938751, 8.760640546314259963826187958248, 10.05872305979619281384375607211, 11.19126087861432537615466177610, 11.67818493311529937614895976665, 12.57527265480265369405726296857

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