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Properties

Label 2-170-85.13-c2-0-14
Degree $2$
Conductor $170$
Sign $0.968 + 0.247i$
Analytic cond. $4.63216$
Root an. cond. $2.15224$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s − 0.458i·3-s + 2i·4-s + (0.150 − 4.99i)5-s + (0.458 − 0.458i)6-s − 6.37i·7-s + (−2 + 2i)8-s + 8.79·9-s + (5.14 − 4.84i)10-s + (9.20 − 9.20i)11-s + 0.916·12-s + (11.6 + 11.6i)13-s + (6.37 − 6.37i)14-s + (−2.28 − 0.0689i)15-s − 4·16-s + (−16.6 + 3.55i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s − 0.152i·3-s + 0.5i·4-s + (0.0300 − 0.999i)5-s + (0.0763 − 0.0763i)6-s − 0.910i·7-s + (−0.250 + 0.250i)8-s + 0.976·9-s + (0.514 − 0.484i)10-s + (0.837 − 0.837i)11-s + 0.0763·12-s + (0.899 + 0.899i)13-s + (0.455 − 0.455i)14-s + (−0.152 − 0.00459i)15-s − 0.250·16-s + (−0.977 + 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.968 + 0.247i$
Analytic conductor: \(4.63216\)
Root analytic conductor: \(2.15224\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1),\ 0.968 + 0.247i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.01516 - 0.253663i\)
\(L(\frac12)\) \(\approx\) \(2.01516 - 0.253663i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 + (-0.150 + 4.99i)T \)
17 \( 1 + (16.6 - 3.55i)T \)
good3 \( 1 + 0.458iT - 9T^{2} \)
7 \( 1 + 6.37iT - 49T^{2} \)
11 \( 1 + (-9.20 + 9.20i)T - 121iT^{2} \)
13 \( 1 + (-11.6 - 11.6i)T + 169iT^{2} \)
19 \( 1 + 14.9T + 361T^{2} \)
23 \( 1 - 20.6T + 529T^{2} \)
29 \( 1 + (-10.1 + 10.1i)T - 841iT^{2} \)
31 \( 1 + (-17.4 - 17.4i)T + 961iT^{2} \)
37 \( 1 + 44.2T + 1.36e3T^{2} \)
41 \( 1 + (4.45 - 4.45i)T - 1.68e3iT^{2} \)
43 \( 1 + (-17.4 + 17.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (39.0 - 39.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (54.4 - 54.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 85.0T + 3.48e3T^{2} \)
61 \( 1 + (54.5 - 54.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (11.8 - 11.8i)T - 4.48e3iT^{2} \)
71 \( 1 + (38.2 + 38.2i)T + 5.04e3iT^{2} \)
73 \( 1 - 104. iT - 5.32e3T^{2} \)
79 \( 1 + (36.6 + 36.6i)T + 6.24e3iT^{2} \)
83 \( 1 + (16.2 - 16.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 126. iT - 7.92e3T^{2} \)
97 \( 1 + 63.7T + 9.40e3T^{2} \)
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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

−12.80167335225367808630975646737, −11.68977311427526113319946029167, −10.63402287532824908448011444901, −9.135657072702333922705096771018, −8.426978430280135316568657949827, −7.02126491916015296261846811823, −6.24398374046914750295356233593, −4.57567611517374818284257498995, −3.92223978616525318406368395862, −1.30363403160145588884809582460, 1.98910585855986735012097194192, 3.42337865064008222240146419154, 4.70191629647420141807023876864, 6.20172159776561711671892813717, 7.03332621299564623785416194061, 8.696920081453714644691188695395, 9.820919225239210266181370912689, 10.68561521827901569625012388863, 11.56131130630184206924129435127, 12.61185127618001987200938621711

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