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Properties

Label 2-170-85.78-c1-0-8
Degree $2$
Conductor $170$
Sign $-0.923 + 0.383i$
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 + (−0.641 − 3.22i)3-s + (−0.707 + 0.707i)4-s + (1.65 − 1.50i)5-s + (−2.73 + 1.82i)6-s + (2.05 − 1.37i)7-s + (0.923 + 0.382i)8-s + (−7.20 + 2.98i)9-s + (−2.02 − 0.948i)10-s + (2.38 + 3.56i)11-s + (2.73 + 1.82i)12-s − 1.04·13-s + (−2.05 − 1.37i)14-s + (−5.91 − 4.35i)15-s − i·16-s + (−2.75 + 3.07i)17-s + ⋯
L(s)  = 1  + (−0.270 − 0.653i)2-s + (−0.370 − 1.86i)3-s + (−0.353 + 0.353i)4-s + (0.738 − 0.674i)5-s + (−1.11 + 0.745i)6-s + (0.775 − 0.517i)7-s + (0.326 + 0.135i)8-s + (−2.40 + 0.995i)9-s + (−0.640 − 0.299i)10-s + (0.719 + 1.07i)11-s + (0.788 + 0.527i)12-s − 0.288·13-s + (−0.548 − 0.366i)14-s + (−1.52 − 1.12i)15-s − 0.250i·16-s + (−0.666 + 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.923 + 0.383i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.195295 - 0.978645i\)
\(L(\frac12)\) \(\approx\) \(0.195295 - 0.978645i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 + 1.50i)T \)
17 \( 1 + (2.75 - 3.07i)T \)
good3 \( 1 + (0.641 + 3.22i)T + (-2.77 + 1.14i)T^{2} \)
7 \( 1 + (-2.05 + 1.37i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-2.38 - 3.56i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + 1.04T + 13T^{2} \)
19 \( 1 + (-2.96 - 1.22i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.740 - 0.147i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (0.174 - 0.0346i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + (-4.06 + 6.08i)T + (-11.8 - 28.6i)T^{2} \)
37 \( 1 + (-1.26 + 0.251i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (5.98 + 1.19i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (-1.34 + 3.23i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 1.87iT - 47T^{2} \)
53 \( 1 + (3.21 - 1.33i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.48 - 8.40i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.70 - 8.57i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (11.5 + 11.5i)T + 67iT^{2} \)
71 \( 1 + (7.90 + 5.27i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (-8.39 - 5.61i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-4.62 + 3.09i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (2.31 + 5.58i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-1.46 + 1.46i)T - 89iT^{2} \)
97 \( 1 + (-1.17 - 0.783i)T + (37.1 + 89.6i)T^{2} \)
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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

−12.23937023410902639819225125020, −11.74876468552804264327484689269, −10.53453356634139647897958433116, −9.199778780610872481208204875753, −8.106982845057000466967794536457, −7.23434215994791422074220666440, −6.05919928356877526553852254612, −4.66129269111144776616271623559, −2.12808352194779655437299175381, −1.25848386310597170304378827332, 3.15696727534628172522206892395, 4.76652169765321532898771971383, 5.55213147965072981503297796269, 6.60700014141122652724350933555, 8.504131061442100329020107656311, 9.252742453780682855233538340233, 10.04900614718767178817435277608, 11.08815811697000753819761375115, 11.61467264628804255523754410223, 13.82574540503384696493773180115

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