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Properties

Label 2-170-17.15-c1-0-4
Degree $2$
Conductor $170$
Sign $0.469 + 0.883i$
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 + (−0.179 − 0.433i)3-s + 1.00i·4-s + (0.923 − 0.382i)5-s + (−0.179 + 0.433i)6-s + (1.29 + 0.536i)7-s + (0.707 − 0.707i)8-s + (1.96 − 1.96i)9-s + (−0.923 − 0.382i)10-s + (−0.117 + 0.284i)11-s + (0.433 − 0.179i)12-s − 5.87i·13-s + (−0.536 − 1.29i)14-s + (−0.331 − 0.331i)15-s − 1.00·16-s + (−4.06 + 0.704i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.103 − 0.249i)3-s + 0.500i·4-s + (0.413 − 0.171i)5-s + (−0.0732 + 0.176i)6-s + (0.489 + 0.202i)7-s + (0.250 − 0.250i)8-s + (0.655 − 0.655i)9-s + (−0.292 − 0.121i)10-s + (−0.0355 + 0.0857i)11-s + (0.124 − 0.0517i)12-s − 1.63i·13-s + (−0.143 − 0.346i)14-s + (−0.0855 − 0.0855i)15-s − 0.250·16-s + (−0.985 + 0.170i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.469 + 0.883i$
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: \((2,\ 170,\ (\ :1/2),\ 0.469 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848035 - 0.509642i\)
\(L(\frac12)\) \(\approx\) \(0.848035 - 0.509642i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.06 - 0.704i)T \)
good3 \( 1 + (0.179 + 0.433i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (-1.29 - 0.536i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.117 - 0.284i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 5.87iT - 13T^{2} \)
19 \( 1 + (-5.56 - 5.56i)T + 19iT^{2} \)
23 \( 1 + (-2.50 + 6.04i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (6.96 - 2.88i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-2.43 - 5.88i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-3.56 - 8.61i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.798 - 0.330i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (6.44 - 6.44i)T - 43iT^{2} \)
47 \( 1 - 6.44iT - 47T^{2} \)
53 \( 1 + (1.88 + 1.88i)T + 53iT^{2} \)
59 \( 1 + (0.396 - 0.396i)T - 59iT^{2} \)
61 \( 1 + (2.21 + 0.917i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 7.06T + 67T^{2} \)
71 \( 1 + (-0.522 - 1.26i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-3.49 + 1.44i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (0.547 - 1.32i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (4.39 + 4.39i)T + 83iT^{2} \)
89 \( 1 + 12.1iT - 89T^{2} \)
97 \( 1 + (0.700 - 0.290i)T + (68.5 - 68.5i)T^{2} \)
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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

−12.63851063382697053615830711708, −11.61447421204031014312999599381, −10.48617066567731745761866655927, −9.731567793336311744759148189759, −8.562078710396895076166692948624, −7.61822703336912300077069890378, −6.28658229009070506689144476199, −4.89343018350643371475343202378, −3.17664233248415959174484423037, −1.37347152451750215933695403994, 1.94445789101437609374854236863, 4.32349759441871362163288438885, 5.41321104790368896182670072061, 6.89542037180136095345351491224, 7.57281524860528372886022251685, 9.146548120771709805339447269554, 9.620165469255193000328167287452, 11.04271706332598091035554052948, 11.47796209827119807711489280600, 13.43606967034343528938386326282

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