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Properties

Label 2-170-17.9-c1-0-2
Degree $2$
Conductor $170$
Sign $0.952 + 0.304i$
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 + (1.93 + 0.802i)3-s − 1.00i·4-s + (−0.382 + 0.923i)5-s + (1.93 − 0.802i)6-s + (−0.434 − 1.05i)7-s + (−0.707 − 0.707i)8-s + (0.985 + 0.985i)9-s + (0.382 + 0.923i)10-s + (−0.233 + 0.0965i)11-s + (0.802 − 1.93i)12-s + 3.10i·13-s + (−1.05 − 0.434i)14-s + (−1.48 + 1.48i)15-s − 1.00·16-s + (−3.02 − 2.80i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (1.11 + 0.463i)3-s − 0.500i·4-s + (−0.171 + 0.413i)5-s + (0.790 − 0.327i)6-s + (−0.164 − 0.396i)7-s + (−0.250 − 0.250i)8-s + (0.328 + 0.328i)9-s + (0.121 + 0.292i)10-s + (−0.0702 + 0.0291i)11-s + (0.231 − 0.559i)12-s + 0.862i·13-s + (−0.280 − 0.116i)14-s + (−0.382 + 0.382i)15-s − 0.250·16-s + (−0.732 − 0.680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.952 + 0.304i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.952 + 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81473 - 0.282605i\)
\(L(\frac12)\) \(\approx\) \(1.81473 - 0.282605i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.02 + 2.80i)T \)
good3 \( 1 + (-1.93 - 0.802i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (0.434 + 1.05i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.233 - 0.0965i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 3.10iT - 13T^{2} \)
19 \( 1 + (0.972 - 0.972i)T - 19iT^{2} \)
23 \( 1 + (0.694 - 0.287i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.432 - 1.04i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.863 + 0.357i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-4.54 - 1.88i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.67 + 8.88i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-5.13 - 5.13i)T + 43iT^{2} \)
47 \( 1 - 9.71iT - 47T^{2} \)
53 \( 1 + (9.25 - 9.25i)T - 53iT^{2} \)
59 \( 1 + (-1.67 - 1.67i)T + 59iT^{2} \)
61 \( 1 + (2.81 + 6.79i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 15.8T + 67T^{2} \)
71 \( 1 + (-2.35 - 0.974i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.80 - 9.19i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-12.9 + 5.36i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-11.0 + 11.0i)T - 83iT^{2} \)
89 \( 1 + 11.9iT - 89T^{2} \)
97 \( 1 + (-6.02 + 14.5i)T + (-68.5 - 68.5i)T^{2} \)
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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

−12.88721880707690877142840811758, −11.67460281863771754751947506989, −10.76102434290887674250997453055, −9.668329347416830075234574894804, −8.934111893117512621501663123267, −7.59102244266885618566493884630, −6.35199243576033645179004948724, −4.55395952208252708012348950479, −3.58788220820314775174632029604, −2.38634083430198361503728871574, 2.39195834083301121507721890043, 3.73163972391245429930786898750, 5.24401821749980172887664092136, 6.54941266561700078508041827140, 7.85484260539115490660584933033, 8.424166220242950947572358157823, 9.405802283909479601019725971207, 10.94899935042592921543633443131, 12.30089859225612552010976455771, 13.06395958489030989287512171435

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