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Properties

Label 2-170-85.73-c1-0-2
Degree $2$
Conductor $170$
Sign $0.996 - 0.0774i$
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 + (−0.241 + 1.21i)3-s + (0.707 + 0.707i)4-s + (1.17 − 1.90i)5-s + (0.688 − 1.03i)6-s + (−0.590 − 0.394i)7-s + (−0.382 − 0.923i)8-s + (1.35 + 0.560i)9-s + (−1.81 + 1.31i)10-s + (3.89 + 2.59i)11-s + (−1.03 + 0.688i)12-s + 0.327i·13-s + (0.394 + 0.590i)14-s + (2.03 + 1.88i)15-s + i·16-s + (4.06 − 0.692i)17-s + ⋯
L(s)  = 1  + (−0.653 − 0.270i)2-s + (−0.139 + 0.701i)3-s + (0.353 + 0.353i)4-s + (0.523 − 0.851i)5-s + (0.281 − 0.420i)6-s + (−0.223 − 0.149i)7-s + (−0.135 − 0.326i)8-s + (0.451 + 0.186i)9-s + (−0.572 + 0.414i)10-s + (1.17 + 0.783i)11-s + (−0.297 + 0.198i)12-s + 0.0907i·13-s + (0.105 + 0.157i)14-s + (0.524 + 0.486i)15-s + 0.250i·16-s + (0.985 − 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.996 - 0.0774i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.996 - 0.0774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946540 + 0.0366936i\)
\(L(\frac12)\) \(\approx\) \(0.946540 + 0.0366936i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.17 + 1.90i)T \)
17 \( 1 + (-4.06 + 0.692i)T \)
good3 \( 1 + (0.241 - 1.21i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (0.590 + 0.394i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (-3.89 - 2.59i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 - 0.327iT - 13T^{2} \)
19 \( 1 + (0.470 - 0.194i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.729 - 0.145i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-1.37 + 6.92i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (2.70 - 1.80i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (10.7 + 2.13i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-1.39 - 7.02i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (0.310 - 0.128i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 4.22T + 47T^{2} \)
53 \( 1 + (1.65 - 4.00i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.52 + 3.68i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (6.63 - 1.32i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + (-0.274 - 0.274i)T + 67iT^{2} \)
71 \( 1 + (8.70 + 13.0i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-9.94 + 6.64i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (5.34 - 8.00i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (-11.5 - 4.80i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (0.374 + 0.374i)T + 89iT^{2} \)
97 \( 1 + (8.19 - 5.47i)T + (37.1 - 89.6i)T^{2} \)
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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

−12.52649793037023353024901166218, −11.81842422927279348698853887751, −10.44093335680688354881527428375, −9.707399212406890931823523175554, −9.142493488332860045233286794726, −7.80466030150185612579109407197, −6.49538117457839112771322077006, −4.99790493267865605859309228564, −3.84043886193528329075302024374, −1.62433583401469915532611057525, 1.54465893450742814058235638604, 3.40798768375358993375102390330, 5.73969426085740280208612603948, 6.61047670279395472817746024063, 7.33787891701068470260150560972, 8.696752360211176089958718902841, 9.724432227743739317595670213015, 10.64246553823529235349648068963, 11.72320834599013140422261682641, 12.66185302588780543650067915162

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