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Properties

Label 2-170-85.59-c1-0-6
Degree $2$
Conductor $170$
Sign $0.734 + 0.678i$
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 + (1.99 + 0.826i)3-s + 1.00i·4-s + (0.399 − 2.20i)5-s + (−0.826 − 1.99i)6-s + (−1.32 − 3.20i)7-s + (0.707 − 0.707i)8-s + (1.18 + 1.18i)9-s + (−1.83 + 1.27i)10-s + (1.63 + 3.94i)11-s + (−0.826 + 1.99i)12-s + 5.21·13-s + (−1.32 + 3.20i)14-s + (2.61 − 4.06i)15-s − 1.00·16-s + (−2.46 + 3.30i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (1.15 + 0.477i)3-s + 0.500i·4-s + (0.178 − 0.983i)5-s + (−0.337 − 0.814i)6-s + (−0.501 − 1.21i)7-s + (0.250 − 0.250i)8-s + (0.393 + 0.393i)9-s + (−0.581 + 0.402i)10-s + (0.492 + 1.18i)11-s + (−0.238 + 0.576i)12-s + 1.44·13-s + (−0.354 + 0.856i)14-s + (0.675 − 1.04i)15-s − 0.250·16-s + (−0.598 + 0.800i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.734 + 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17357 - 0.459002i\)
\(L(\frac12)\) \(\approx\) \(1.17357 - 0.459002i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.399 + 2.20i)T \)
17 \( 1 + (2.46 - 3.30i)T \)
good3 \( 1 + (-1.99 - 0.826i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (1.32 + 3.20i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.63 - 3.94i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 5.21T + 13T^{2} \)
19 \( 1 + (0.305 - 0.305i)T - 19iT^{2} \)
23 \( 1 + (6.86 - 2.84i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-3.46 - 1.43i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (0.648 - 1.56i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-4.56 - 1.89i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.20 - 1.74i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (2.38 - 2.38i)T - 43iT^{2} \)
47 \( 1 + 3.47T + 47T^{2} \)
53 \( 1 + (9.73 + 9.73i)T + 53iT^{2} \)
59 \( 1 + (-6.37 - 6.37i)T + 59iT^{2} \)
61 \( 1 + (-5.75 + 2.38i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 0.409iT - 67T^{2} \)
71 \( 1 + (3.84 - 9.28i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (0.401 - 0.970i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.218 - 0.527i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \)
89 \( 1 + 14.5iT - 89T^{2} \)
97 \( 1 + (-3.70 + 8.95i)T + (-68.5 - 68.5i)T^{2} \)
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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

−12.88574939505877259978032453283, −11.60024766814910690650464595193, −10.19225379663555196040441261714, −9.701837896124123215718760420000, −8.678571336510865656562415858002, −7.988676292929203685509639345588, −6.49868011357483322585891784518, −4.29620628133978886838141936811, −3.66339821512619013699682476077, −1.65104172801805019404873629134, 2.29535402048540539684245100484, 3.42339239441262041066451107403, 5.96347458115859837422405532776, 6.52924424803448327398836520053, 7.983785169412895937066593770908, 8.709346724272843170622566963067, 9.437935593869684774136679149471, 10.81732488798401328861846178199, 11.78606683577016574093908667159, 13.37135195144749828459008404848

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