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Properties

Label 2-170-85.59-c1-0-7
Degree $2$
Conductor $170$
Sign $0.583 - 0.812i$
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 + (−2.03 − 0.925i)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 + (−0.784 − 2.09i)10-s + (−0.543 − 1.31i)11-s + (−1.18 + 2.86i)12-s − 3.10·13-s + (1.09 − 2.64i)14-s + (−4.73 − 5.06i)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 + (−0.910 − 0.414i)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 + (−0.248 − 0.662i)10-s + (−0.163 − 0.395i)11-s + (−0.342 + 0.827i)12-s − 0.862·13-s + (0.292 − 0.705i)14-s + (−1.22 − 1.30i)15-s − 0.250·16-s + (0.268 − 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.583 - 0.812i$
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: \((2,\ 170,\ (\ :1/2),\ 0.583 - 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72118 + 0.883072i\)
\(L(\frac12)\) \(\approx\) \(1.72118 + 0.883072i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 0.925i)T \)
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

−13.19399262214262293842702274193, −12.32303288039948601741051706529, −10.72898138728440968848116739881, −9.719903260497484464065678289271, −8.596012151390455259848732724500, −7.87236551222388728818159343314, −6.99315148303854502197094627269, −4.79772698694070085132915984614, −3.95898469745518878298623830286, −2.97467170486455880584687541958, 2.32408058725325788087142809636, 3.09854397635189925826162642511, 4.45100391384730490803493393916, 6.50137877875638399033946648032, 7.50865652698452531014143830565, 8.584265138516201068615288526267, 9.372449403381124537858133187112, 10.68333743803782752754430617680, 12.11993992669451782374241688356, 12.62940462227064357897221881641

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