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Properties

Label 2-170-85.63-c1-0-3
Degree $2$
Conductor $170$
Sign $0.990 - 0.136i$
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 + (1.87 + 0.372i)3-s + (0.707 + 0.707i)4-s + (2.15 + 0.598i)5-s + (−1.58 − 1.06i)6-s + (−0.622 + 0.930i)7-s + (−0.382 − 0.923i)8-s + (0.596 + 0.247i)9-s + (−1.76 − 1.37i)10-s + (−1.86 + 2.79i)11-s + (1.06 + 1.58i)12-s − 1.47i·13-s + (0.930 − 0.622i)14-s + (3.81 + 1.92i)15-s + i·16-s + (−1.32 − 3.90i)17-s + ⋯
L(s)  = 1  + (−0.653 − 0.270i)2-s + (1.08 + 0.215i)3-s + (0.353 + 0.353i)4-s + (0.963 + 0.267i)5-s + (−0.648 − 0.433i)6-s + (−0.235 + 0.351i)7-s + (−0.135 − 0.326i)8-s + (0.198 + 0.0823i)9-s + (−0.556 − 0.435i)10-s + (−0.562 + 0.842i)11-s + (0.306 + 0.458i)12-s − 0.409i·13-s + (0.248 − 0.166i)14-s + (0.984 + 0.496i)15-s + 0.250i·16-s + (−0.322 − 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.990 - 0.136i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.990 - 0.136i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25876 + 0.0860581i\)
\(L(\frac12)\) \(\approx\) \(1.25876 + 0.0860581i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.15 - 0.598i)T \)
17 \( 1 + (1.32 + 3.90i)T \)
good3 \( 1 + (-1.87 - 0.372i)T + (2.77 + 1.14i)T^{2} \)
7 \( 1 + (0.622 - 0.930i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (1.86 - 2.79i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + 1.47iT - 13T^{2} \)
19 \( 1 + (-4.07 + 1.68i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.581 + 2.92i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (3.34 + 0.664i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (-2.94 - 4.41i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (-1.64 + 8.28i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (10.6 - 2.11i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (6.16 - 2.55i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 6.49T + 47T^{2} \)
53 \( 1 + (-2.17 + 5.24i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.784 - 1.89i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.266 - 1.34i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-7.14 - 7.14i)T + 67iT^{2} \)
71 \( 1 + (-1.30 + 0.874i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (7.04 + 10.5i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (-6.00 - 4.01i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (-11.8 - 4.89i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-5.20 - 5.20i)T + 89iT^{2} \)
97 \( 1 + (-5.44 - 8.14i)T + (-37.1 + 89.6i)T^{2} \)
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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

−12.94456232829827891810435690325, −11.71621661012759302215191623638, −10.39980373786681912627859971550, −9.620724475526637830305113926125, −9.011411778747141209309934433870, −7.86581869169026504624200404799, −6.73143505583511918542407372713, −5.17117931160898426208969839674, −3.13507608942478595171298244970, −2.24761558312362947868333795167, 1.81726282335956608150348142472, 3.30441471192640873470057969898, 5.41247714773798618205666687723, 6.56262653255940539949994276625, 7.907658590838012388556466920790, 8.612868522896043767337321832566, 9.568604923766002229221822143497, 10.36421382288152256529865577530, 11.63990601590092390484534726477, 13.33385021161980096220278801612

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