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Properties

Label 2-170-85.23-c1-0-5
Degree $2$
Conductor $170$
Sign $0.988 - 0.151i$
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 + (2.12 + 1.42i)3-s + (0.707 − 0.707i)4-s + (−2.23 − 0.143i)5-s + (2.51 + 0.499i)6-s + (0.0411 + 0.00818i)7-s + (0.382 − 0.923i)8-s + (1.36 + 3.29i)9-s + (−2.11 + 0.721i)10-s + (0.476 − 2.39i)11-s + (2.51 − 0.499i)12-s − 2.17·13-s + (0.0411 − 0.00818i)14-s + (−4.54 − 3.48i)15-s − i·16-s + (−2.37 + 3.36i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.270i)2-s + (1.22 + 0.821i)3-s + (0.353 − 0.353i)4-s + (−0.997 − 0.0641i)5-s + (1.02 + 0.204i)6-s + (0.0155 + 0.00309i)7-s + (0.135 − 0.326i)8-s + (0.454 + 1.09i)9-s + (−0.669 + 0.228i)10-s + (0.143 − 0.721i)11-s + (0.725 − 0.144i)12-s − 0.601·13-s + (0.0109 − 0.00218i)14-s + (−1.17 − 0.898i)15-s − 0.250i·16-s + (−0.576 + 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.988 - 0.151i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.988 - 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93476 + 0.147100i\)
\(L(\frac12)\) \(\approx\) \(1.93476 + 0.147100i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.143i)T \)
17 \( 1 + (2.37 - 3.36i)T \)
good3 \( 1 + (-2.12 - 1.42i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (-0.0411 - 0.00818i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.476 + 2.39i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + 2.17T + 13T^{2} \)
19 \( 1 + (1.43 - 3.45i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (3.96 + 5.93i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (-4.73 + 7.08i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-0.226 - 1.13i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (4.07 - 6.10i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-0.400 - 0.599i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-8.37 - 3.46i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 4.13iT - 47T^{2} \)
53 \( 1 + (-2.01 - 4.87i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (8.77 - 3.63i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.87 + 3.92i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (-11.0 - 11.0i)T + 67iT^{2} \)
71 \( 1 + (-5.22 + 1.04i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-11.1 + 2.22i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (7.04 + 1.40i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (15.6 - 6.46i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-11.5 + 11.5i)T - 89iT^{2} \)
97 \( 1 + (4.02 - 0.800i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.81669704806915847009194528328, −11.90879950409485777904927598372, −10.76667988081305208726091123763, −9.895970771608110100144004313906, −8.566827287395699837864937156444, −7.996890772357709130737943046837, −6.34533806430806052613441572567, −4.52993845199125110254597674053, −3.84911050443721949737153247791, −2.65118393849233521210300819183, 2.32301086331336856098800467278, 3.58841560708817857154007292671, 4.88670167326713772468236268647, 6.92085657634041811333230098568, 7.36497511940410871927835025563, 8.366161723227954621495867300566, 9.391977196999012532356303142969, 11.07867010193489487337932717850, 12.18523200580460069608672578784, 12.77070229674876167315792005057

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