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Properties

Label 2-170-85.59-c1-0-2
Degree $2$
Conductor $170$
Sign $0.998 - 0.0595i$
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 + (−0.857 − 0.355i)3-s + 1.00i·4-s + (1.64 + 1.51i)5-s + (0.355 + 0.857i)6-s + (0.939 + 2.26i)7-s + (0.707 − 0.707i)8-s + (−1.51 − 1.51i)9-s + (−0.0908 − 2.23i)10-s + (1.42 + 3.44i)11-s + (0.355 − 0.857i)12-s + 4.92·13-s + (0.939 − 2.26i)14-s + (−0.871 − 1.88i)15-s − 1.00·16-s + (3.07 − 2.74i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.494 − 0.204i)3-s + 0.500i·4-s + (0.735 + 0.677i)5-s + (0.144 + 0.349i)6-s + (0.355 + 0.857i)7-s + (0.250 − 0.250i)8-s + (−0.504 − 0.504i)9-s + (−0.0287 − 0.706i)10-s + (0.429 + 1.03i)11-s + (0.102 − 0.247i)12-s + 1.36·13-s + (0.251 − 0.606i)14-s + (−0.224 − 0.486i)15-s − 0.250·16-s + (0.746 − 0.665i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.998 - 0.0595i$
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: \((2,\ 170,\ (\ :1/2),\ 0.998 - 0.0595i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.902132 + 0.0268691i\)
\(L(\frac12)\) \(\approx\) \(0.902132 + 0.0268691i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 1.51i)T \)
17 \( 1 + (-3.07 + 2.74i)T \)
good3 \( 1 + (0.857 + 0.355i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-0.939 - 2.26i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.42 - 3.44i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 4.92T + 13T^{2} \)
19 \( 1 + (-2.88 + 2.88i)T - 19iT^{2} \)
23 \( 1 + (4.93 - 2.04i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (9.69 + 4.01i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (2.21 - 5.35i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.79 - 0.743i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-6.01 + 2.48i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-1.23 + 1.23i)T - 43iT^{2} \)
47 \( 1 + 7.40T + 47T^{2} \)
53 \( 1 + (3.36 + 3.36i)T + 53iT^{2} \)
59 \( 1 + (3.52 + 3.52i)T + 59iT^{2} \)
61 \( 1 + (2.70 - 1.11i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 + (-4.66 + 11.2i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-2.67 + 6.44i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.57 - 6.22i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (11.1 + 11.1i)T + 83iT^{2} \)
89 \( 1 - 8.04iT - 89T^{2} \)
97 \( 1 + (-2.77 + 6.70i)T + (-68.5 - 68.5i)T^{2} \)
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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

−12.46933742416692610124483814417, −11.61735980386725028129112179483, −11.00373454953970807137972452471, −9.645747817500080866758619245448, −9.088189388966851004853955424368, −7.60743881512092826754712539032, −6.37933490693995244702110767508, −5.42696823237117383398720238129, −3.37815202883649086777355642769, −1.81329048243711988710759207025, 1.28757023724882881265976420922, 3.99414121134324540224596837154, 5.65445748631573311323275974218, 6.02187795396267132866001641603, 7.81578818318349279117741920234, 8.571973414161663656165188664626, 9.728139403882433819185607899933, 10.76310670556436479806516247453, 11.39584223583571178366450524918, 12.94802142339817513814013785421

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