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Properties

Label 2-170-17.8-c1-0-4
Degree $2$
Conductor $170$
Sign $0.0465 + 0.998i$
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 + (0.548 − 1.32i)3-s − 1.00i·4-s + (−0.923 − 0.382i)5-s + (−0.548 − 1.32i)6-s + (0.599 − 0.248i)7-s + (−0.707 − 0.707i)8-s + (0.668 + 0.668i)9-s + (−0.923 + 0.382i)10-s + (−1.42 − 3.44i)11-s + (−1.32 − 0.548i)12-s + 4.08i·13-s + (0.248 − 0.599i)14-s + (−1.01 + 1.01i)15-s − 1.00·16-s + (3.85 − 1.46i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.316 − 0.764i)3-s − 0.500i·4-s + (−0.413 − 0.171i)5-s + (−0.223 − 0.540i)6-s + (0.226 − 0.0938i)7-s + (−0.250 − 0.250i)8-s + (0.222 + 0.222i)9-s + (−0.292 + 0.121i)10-s + (−0.430 − 1.03i)11-s + (−0.382 − 0.158i)12-s + 1.13i·13-s + (0.0663 − 0.160i)14-s + (−0.261 + 0.261i)15-s − 0.250·16-s + (0.934 − 0.355i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0465 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0465 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.0465 + 0.998i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.0465 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10713 - 1.05678i\)
\(L(\frac12)\) \(\approx\) \(1.10713 - 1.05678i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 + (-3.85 + 1.46i)T \)
good3 \( 1 + (-0.548 + 1.32i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-0.599 + 0.248i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.42 + 3.44i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 4.08iT - 13T^{2} \)
19 \( 1 + (3.03 - 3.03i)T - 19iT^{2} \)
23 \( 1 + (-2.78 - 6.72i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.66 + 0.689i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.90 + 4.59i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (3.12 - 7.53i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.78 + 0.741i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-0.740 - 0.740i)T + 43iT^{2} \)
47 \( 1 + 4.71iT - 47T^{2} \)
53 \( 1 + (6.16 - 6.16i)T - 53iT^{2} \)
59 \( 1 + (7.34 + 7.34i)T + 59iT^{2} \)
61 \( 1 + (3.27 - 1.35i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 3.48T + 67T^{2} \)
71 \( 1 + (-5.07 + 12.2i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.53 + 2.29i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.112 + 0.270i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-1.75 + 1.75i)T - 83iT^{2} \)
89 \( 1 + 3.59iT - 89T^{2} \)
97 \( 1 + (17.7 + 7.36i)T + (68.5 + 68.5i)T^{2} \)
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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

−12.55271153895457310923855725092, −11.69286028915182536416929273136, −10.84582041337180506676949036265, −9.560656577282205680097449471404, −8.269587119604494542894776168238, −7.40254991183300901615098209203, −6.05836456337955610191072753841, −4.67418787141383099183882738768, −3.26036812878927516730399527464, −1.58224692028146268308300354371, 2.96279971576369389063124726206, 4.25460500075464692779378001458, 5.19649055702999579904840990838, 6.74163706442464675284642451204, 7.80686744573673138497561288287, 8.829405961520166989144398251108, 10.09775906776907138765013137354, 10.86018268702649294062149407954, 12.46270081866087440212470524628, 12.76997798903342567139247886522

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