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Properties

Label 2-170-85.7-c1-0-1
Degree $2$
Conductor $170$
Sign $-0.206 - 0.978i$
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  + (−0.923 + 0.382i)2-s + (0.491 + 2.47i)3-s + (0.707 − 0.707i)4-s + (0.780 + 2.09i)5-s + (−1.39 − 2.09i)6-s + (3.89 − 2.60i)7-s + (−0.382 + 0.923i)8-s + (−3.09 + 1.28i)9-s + (−1.52 − 1.63i)10-s + (−4.03 + 2.69i)11-s + (2.09 + 1.39i)12-s − 0.696i·13-s + (−2.60 + 3.89i)14-s + (−4.79 + 2.95i)15-s − i·16-s + (−1.26 − 3.92i)17-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (0.283 + 1.42i)3-s + (0.353 − 0.353i)4-s + (0.349 + 0.937i)5-s + (−0.571 − 0.855i)6-s + (1.47 − 0.984i)7-s + (−0.135 + 0.326i)8-s + (−1.03 + 0.426i)9-s + (−0.481 − 0.517i)10-s + (−1.21 + 0.812i)11-s + (0.604 + 0.404i)12-s − 0.193i·13-s + (−0.696 + 1.04i)14-s + (−1.23 + 0.763i)15-s − 0.250i·16-s + (−0.306 − 0.952i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.206 - 0.978i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656984 + 0.809849i\)
\(L(\frac12)\) \(\approx\) \(0.656984 + 0.809849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (-0.780 - 2.09i)T \)
17 \( 1 + (1.26 + 3.92i)T \)
good3 \( 1 + (-0.491 - 2.47i)T + (-2.77 + 1.14i)T^{2} \)
7 \( 1 + (-3.89 + 2.60i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (4.03 - 2.69i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + 0.696iT - 13T^{2} \)
19 \( 1 + (3.16 + 1.31i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.49 - 1.09i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (1.47 + 7.40i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (-1.62 - 1.08i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (-2.02 + 0.403i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (-1.40 + 7.04i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (-4.22 - 1.74i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 2.85T + 47T^{2} \)
53 \( 1 + (-0.469 - 1.13i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.91 - 9.44i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (4.67 + 0.930i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (0.385 - 0.385i)T - 67iT^{2} \)
71 \( 1 + (0.365 - 0.546i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (0.966 + 0.645i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-1.73 - 2.60i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (-8.40 + 3.48i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (6.89 - 6.89i)T - 89iT^{2} \)
97 \( 1 + (-1.41 - 0.945i)T + (37.1 + 89.6i)T^{2} \)
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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

−13.42853024114121439744449605245, −11.36311374575753945955486915825, −10.70031287296979126685956070031, −10.21068284418997728180148234982, −9.212432609338191322783181858660, −7.898779341378382019962258142485, −7.12538751451434919266495066130, −5.26567274900983903009716177510, −4.33906683599045543928099837538, −2.52717133501754954669485991046, 1.41677194834877248236669707436, 2.42611726529594637095675758245, 5.03257694738980225278083041737, 6.19643881856039304854809154640, 7.78021273134846827475632938617, 8.370907738874506580859528201880, 8.912603438773319885343730869538, 10.69911934619483878398199983792, 11.58366152947401600612804318371, 12.69226290265283769450400271291

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