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Properties

Label 2-170-85.59-c1-0-0
Degree $2$
Conductor $170$
Sign $0.526 - 0.850i$
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 + (−2.86 − 1.18i)3-s + 1.00i·4-s + (−2.09 − 0.784i)5-s + (1.18 + 2.86i)6-s + (1.09 + 2.64i)7-s + (0.707 − 0.707i)8-s + (4.67 + 4.67i)9-s + (0.925 + 2.03i)10-s + (−0.543 − 1.31i)11-s + (1.18 − 2.86i)12-s + 3.10·13-s + (1.09 − 2.64i)14-s + (5.06 + 4.73i)15-s − 1.00·16-s + (−1.10 + 3.97i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−1.65 − 0.685i)3-s + 0.500i·4-s + (−0.936 − 0.350i)5-s + (0.484 + 1.16i)6-s + (0.413 + 0.997i)7-s + (0.250 − 0.250i)8-s + (1.55 + 1.55i)9-s + (0.292 + 0.643i)10-s + (−0.163 − 0.395i)11-s + (0.342 − 0.827i)12-s + 0.862·13-s + (0.292 − 0.705i)14-s + (1.30 + 1.22i)15-s − 0.250·16-s + (−0.268 + 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.526 - 0.850i$
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: \((2,\ 170,\ (\ :1/2),\ 0.526 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.239526 + 0.133413i\)
\(L(\frac12)\) \(\approx\) \(0.239526 + 0.133413i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 + 0.784i)T \)
17 \( 1 + (1.10 - 3.97i)T \)
good3 \( 1 + (2.86 + 1.18i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-1.09 - 2.64i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.543 + 1.31i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 3.10T + 13T^{2} \)
19 \( 1 + (5.83 - 5.83i)T - 19iT^{2} \)
23 \( 1 + (4.30 - 1.78i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-4.05 - 1.67i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (0.0583 - 0.140i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (3.11 + 1.28i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.28 - 1.35i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-0.136 + 0.136i)T - 43iT^{2} \)
47 \( 1 + 5.42T + 47T^{2} \)
53 \( 1 + (-6.74 - 6.74i)T + 53iT^{2} \)
59 \( 1 + (-1.97 - 1.97i)T + 59iT^{2} \)
61 \( 1 + (8.95 - 3.70i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 6.86iT - 67T^{2} \)
71 \( 1 + (-1.40 + 3.38i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-4.32 + 10.4i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.23 - 10.2i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (3.18 + 3.18i)T + 83iT^{2} \)
89 \( 1 + 0.619iT - 89T^{2} \)
97 \( 1 + (5.75 - 13.9i)T + (-68.5 - 68.5i)T^{2} \)
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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

−12.22771314077367808784713364125, −12.13245886133170528278186592169, −11.05162161353652843711281273609, −10.47934138979599043499163864867, −8.584622908989321821989297976024, −7.985562134250064813588663308128, −6.47651491853877482066348664995, −5.55699269297822759852795933845, −4.10359717508839150753845546321, −1.59997955089306923871330230965, 0.38390268384251310573464431518, 4.16068987298866126547057739492, 4.88427802460869437558570727818, 6.45865240777440390613143245869, 7.07461338252325226812443431097, 8.422200898126333143645935173648, 9.929849322580384488969104974250, 10.82633346173900777907170876388, 11.18852561939638490130938015138, 12.21360816866927195653590450311

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