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Properties

Label 2-170-85.23-c1-0-3
Degree $2$
Conductor $170$
Sign $0.964 + 0.265i$
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 + (−0.432 − 0.288i)3-s + (0.707 − 0.707i)4-s + (0.601 + 2.15i)5-s + (−0.509 − 0.101i)6-s + (3.10 + 0.618i)7-s + (0.382 − 0.923i)8-s + (−1.04 − 2.52i)9-s + (1.37 + 1.75i)10-s + (0.0548 − 0.275i)11-s + (−0.509 + 0.101i)12-s − 1.43·13-s + (3.10 − 0.618i)14-s + (0.361 − 1.10i)15-s − i·16-s + (−0.813 − 4.04i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.270i)2-s + (−0.249 − 0.166i)3-s + (0.353 − 0.353i)4-s + (0.269 + 0.963i)5-s + (−0.208 − 0.0414i)6-s + (1.17 + 0.233i)7-s + (0.135 − 0.326i)8-s + (−0.348 − 0.840i)9-s + (0.436 + 0.556i)10-s + (0.0165 − 0.0832i)11-s + (−0.147 + 0.0292i)12-s − 0.397·13-s + (0.830 − 0.165i)14-s + (0.0934 − 0.285i)15-s − 0.250i·16-s + (−0.197 − 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.964 + 0.265i$
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: \((2,\ 170,\ (\ :1/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60856 - 0.217592i\)
\(L(\frac12)\) \(\approx\) \(1.60856 - 0.217592i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.601 - 2.15i)T \)
17 \( 1 + (0.813 + 4.04i)T \)
good3 \( 1 + (0.432 + 0.288i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (-3.10 - 0.618i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.0548 + 0.275i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + 1.43T + 13T^{2} \)
19 \( 1 + (3.00 - 7.24i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (2.71 + 4.06i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (2.28 - 3.42i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-1.68 - 8.47i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-4.87 + 7.29i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (4.80 + 7.18i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (6.28 + 2.60i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 2.58iT - 47T^{2} \)
53 \( 1 + (-1.55 - 3.76i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.245 + 0.101i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-7.53 + 5.03i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (-3.18 - 3.18i)T + 67iT^{2} \)
71 \( 1 + (5.40 - 1.07i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (2.68 - 0.534i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-10.5 - 2.09i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (-10.1 + 4.21i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.32 - 5.32i)T - 89iT^{2} \)
97 \( 1 + (-11.7 + 2.34i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.47207408952581991574747780467, −11.82727180659107326912597616892, −10.94353076924044500967325612810, −10.06735899190914111620626746446, −8.623401345465390115673875618900, −7.25559330825100633386902206523, −6.21919181016106411642840214330, −5.16610471221434145480660886724, −3.62125222253638909224170940374, −2.09464698271449980664447784619, 2.06933377965617258342663926022, 4.41432637858455491162088701418, 4.97516791745940857561407060959, 6.13997512596712886895192205612, 7.78661694526639287222739166831, 8.423524232092362711941667104034, 9.867444594579146692739088748729, 11.20924646527548302004685759468, 11.70330845113799809200797121785, 13.20543952523105693342879630366

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