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Properties

Label 2-170-85.19-c1-0-9
Degree $2$
Conductor $170$
Sign $-0.807 + 0.589i$
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  + (0.707 − 0.707i)2-s + (−0.771 − 1.86i)3-s − 1.00i·4-s + (−2.22 + 0.203i)5-s + (−1.86 − 0.771i)6-s + (−2.47 − 1.02i)7-s + (−0.707 − 0.707i)8-s + (−0.750 + 0.750i)9-s + (−1.43 + 1.71i)10-s + (1.21 + 0.502i)11-s + (−1.86 + 0.771i)12-s + 6.94·13-s + (−2.47 + 1.02i)14-s + (2.09 + 3.98i)15-s − 1.00·16-s + (0.815 − 4.04i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.445 − 1.07i)3-s − 0.500i·4-s + (−0.995 + 0.0908i)5-s + (−0.760 − 0.314i)6-s + (−0.935 − 0.387i)7-s + (−0.250 − 0.250i)8-s + (−0.250 + 0.250i)9-s + (−0.452 + 0.543i)10-s + (0.366 + 0.151i)11-s + (−0.537 + 0.222i)12-s + 1.92·13-s + (−0.661 + 0.273i)14-s + (0.541 + 1.03i)15-s − 0.250·16-s + (0.197 − 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.807 + 0.589i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.807 + 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.310729 - 0.952234i\)
\(L(\frac12)\) \(\approx\) \(0.310729 - 0.952234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (2.22 - 0.203i)T \)
17 \( 1 + (-0.815 + 4.04i)T \)
good3 \( 1 + (0.771 + 1.86i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (2.47 + 1.02i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.21 - 0.502i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 6.94T + 13T^{2} \)
19 \( 1 + (3.81 + 3.81i)T + 19iT^{2} \)
23 \( 1 + (0.333 - 0.804i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.562 - 1.35i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-5.75 + 2.38i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-2.67 - 6.46i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.84 - 6.87i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-4.16 - 4.16i)T + 43iT^{2} \)
47 \( 1 - 5.46T + 47T^{2} \)
53 \( 1 + (6.13 - 6.13i)T - 53iT^{2} \)
59 \( 1 + (4.90 - 4.90i)T - 59iT^{2} \)
61 \( 1 + (-2.72 + 6.58i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 6.64iT - 67T^{2} \)
71 \( 1 + (-9.18 + 3.80i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-7.32 + 3.03i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.160 - 0.0663i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (5.79 - 5.79i)T - 83iT^{2} \)
89 \( 1 + 1.81iT - 89T^{2} \)
97 \( 1 + (-5.24 + 2.17i)T + (68.5 - 68.5i)T^{2} \)
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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

−12.40371636061886603924053558656, −11.53794610128445506246225617062, −10.87524825506135482034017467489, −9.426878347558097899495489056865, −8.070210879465894877957280801913, −6.76427883559346909623014920942, −6.27820020936757444911837661092, −4.36863202962361367369687321989, −3.17088969404832357817264240664, −0.918288309605207893812739363574, 3.64806548976663158339049935452, 4.09542174788814016009685965728, 5.70162461629937768021327411610, 6.51342181470487842051588884531, 8.153710741241696519306266478841, 8.963753636425584406048747524103, 10.38696412575961398718812271784, 11.13664381631811618982877770374, 12.26389785588928559811912191834, 13.04183601377882513740240514355

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