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Properties

Label 2-170-85.48-c1-0-8
Degree $2$
Conductor $170$
Sign $-0.0698 + 0.997i$
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  + (0.382 − 0.923i)2-s + (1.02 − 0.683i)3-s + (−0.707 − 0.707i)4-s + (−0.554 − 2.16i)5-s + (−0.239 − 1.20i)6-s + (0.156 − 0.0311i)7-s + (−0.923 + 0.382i)8-s + (−0.569 + 1.37i)9-s + (−2.21 − 0.317i)10-s + (1.18 − 0.235i)11-s + (−1.20 − 0.239i)12-s − 0.898i·13-s + (0.0311 − 0.156i)14-s + (−2.04 − 1.83i)15-s + i·16-s + (3.77 + 1.65i)17-s + ⋯
L(s)  = 1  + (0.270 − 0.653i)2-s + (0.590 − 0.394i)3-s + (−0.353 − 0.353i)4-s + (−0.247 − 0.968i)5-s + (−0.0979 − 0.492i)6-s + (0.0592 − 0.0117i)7-s + (−0.326 + 0.135i)8-s + (−0.189 + 0.458i)9-s + (−0.699 − 0.100i)10-s + (0.356 − 0.0709i)11-s + (−0.348 − 0.0692i)12-s − 0.249i·13-s + (0.00833 − 0.0418i)14-s + (−0.528 − 0.474i)15-s + 0.250i·16-s + (0.916 + 0.400i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0698 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0698 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.0698 + 0.997i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.0698 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.980614 - 1.05164i\)
\(L(\frac12)\) \(\approx\) \(0.980614 - 1.05164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (0.554 + 2.16i)T \)
17 \( 1 + (-3.77 - 1.65i)T \)
good3 \( 1 + (-1.02 + 0.683i)T + (1.14 - 2.77i)T^{2} \)
7 \( 1 + (-0.156 + 0.0311i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (-1.18 + 0.235i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + 0.898iT - 13T^{2} \)
19 \( 1 + (-1.36 - 3.30i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.98 + 4.47i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (0.512 - 0.342i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (6.58 + 1.30i)T + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (-4.96 - 7.43i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-1.80 - 1.20i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.109 + 0.264i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.39T + 47T^{2} \)
53 \( 1 + (-10.4 - 4.33i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (11.0 + 4.58i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.26 - 1.88i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (4.51 + 4.51i)T + 67iT^{2} \)
71 \( 1 + (-0.812 + 4.08i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (9.69 + 1.92i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (1.66 + 8.36i)T + (-72.9 + 30.2i)T^{2} \)
83 \( 1 + (0.0386 - 0.0933i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (4.97 + 4.97i)T + 89iT^{2} \)
97 \( 1 + (-8.62 - 1.71i)T + (89.6 + 37.1i)T^{2} \)
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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

−12.62628550061344545963415495701, −11.74036266696558321675832387275, −10.60946404766247842718177868861, −9.419412676687155814314293211690, −8.454296708430643761225798278645, −7.64412468551321289830279308075, −5.81347133876596209175954244880, −4.63168153452272270273327889360, −3.22143450132865348333292608397, −1.51362578560978538333473450966, 2.99785535631804264857633255000, 3.97382090817736574179152196067, 5.58942550135519856737966856448, 6.86232457828880026271293612052, 7.67776767187231462817517972088, 9.006251502160647047550779207662, 9.742749034009281031311984750401, 11.13414581884189297720521709690, 12.01266658118000257833727800370, 13.34454190157907462229582273771

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