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Properties

Label 2-170-85.19-c1-0-3
Degree $2$
Conductor $170$
Sign $0.999 - 0.0239i$
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 + (0.394 + 0.953i)3-s − 1.00i·4-s + (0.473 + 2.18i)5-s + (0.953 + 0.394i)6-s + (0.363 + 0.150i)7-s + (−0.707 − 0.707i)8-s + (1.36 − 1.36i)9-s + (1.88 + 1.21i)10-s + (2.68 + 1.11i)11-s + (0.953 − 0.394i)12-s − 3.68·13-s + (0.363 − 0.150i)14-s + (−1.89 + 1.31i)15-s − 1.00·16-s + (−2.47 − 3.30i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.227 + 0.550i)3-s − 0.500i·4-s + (0.211 + 0.977i)5-s + (0.389 + 0.161i)6-s + (0.137 + 0.0569i)7-s + (−0.250 − 0.250i)8-s + (0.456 − 0.456i)9-s + (0.594 + 0.382i)10-s + (0.810 + 0.335i)11-s + (0.275 − 0.113i)12-s − 1.02·13-s + (0.0972 − 0.0402i)14-s + (−0.489 + 0.339i)15-s − 0.250·16-s + (−0.599 − 0.800i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.999 - 0.0239i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.999 - 0.0239i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60847 + 0.0192893i\)
\(L(\frac12)\) \(\approx\) \(1.60847 + 0.0192893i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.473 - 2.18i)T \)
17 \( 1 + (2.47 + 3.30i)T \)
good3 \( 1 + (-0.394 - 0.953i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (-0.363 - 0.150i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-2.68 - 1.11i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 3.68T + 13T^{2} \)
19 \( 1 + (1.90 + 1.90i)T + 19iT^{2} \)
23 \( 1 + (0.0427 - 0.103i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.95 + 4.72i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (5.35 - 2.21i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (0.770 + 1.86i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.671 + 1.62i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-8.76 - 8.76i)T + 43iT^{2} \)
47 \( 1 + 6.55T + 47T^{2} \)
53 \( 1 + (-4.22 + 4.22i)T - 53iT^{2} \)
59 \( 1 + (0.866 - 0.866i)T - 59iT^{2} \)
61 \( 1 + (4.08 - 9.86i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 1.14iT - 67T^{2} \)
71 \( 1 + (-3.75 + 1.55i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-12.1 + 5.02i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-12.6 - 5.22i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-5.50 + 5.50i)T - 83iT^{2} \)
89 \( 1 + 5.18iT - 89T^{2} \)
97 \( 1 + (17.4 - 7.22i)T + (68.5 - 68.5i)T^{2} \)
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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

−12.72035960786178869609509993059, −11.71068357714057032270851204555, −10.84821843126007469433713316540, −9.781217870411411749484854599288, −9.236205791641572794210743955517, −7.31194953055120334246178934395, −6.41645118743768707888313100819, −4.82188697601329139959525627173, −3.71605528658424315645118918013, −2.34435512155834332370398515097, 1.89580696937536649676528955778, 4.05171623150235312920649240546, 5.14421461549106635542334091616, 6.41821118448735022490907911273, 7.55154252415257045339287084500, 8.477982573524537958119630830963, 9.481546989512431888087536915418, 10.92713845259465148520717075760, 12.36075029287217155391125828232, 12.70517532928472639798632724313

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