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Properties

Label 2-170-17.2-c1-0-4
Degree $2$
Conductor $170$
Sign $0.890 - 0.454i$
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 + (1.36 − 0.564i)3-s + 1.00i·4-s + (0.382 + 0.923i)5-s + (1.36 + 0.564i)6-s + (1.35 − 3.27i)7-s + (−0.707 + 0.707i)8-s + (−0.581 + 0.581i)9-s + (−0.382 + 0.923i)10-s + (−4.35 − 1.80i)11-s + (0.564 + 1.36i)12-s + 5.47i·13-s + (3.27 − 1.35i)14-s + (1.04 + 1.04i)15-s − 1.00·16-s + (2.52 − 3.25i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.787 − 0.326i)3-s + 0.500i·4-s + (0.171 + 0.413i)5-s + (0.556 + 0.230i)6-s + (0.513 − 1.23i)7-s + (−0.250 + 0.250i)8-s + (−0.193 + 0.193i)9-s + (−0.121 + 0.292i)10-s + (−1.31 − 0.543i)11-s + (0.163 + 0.393i)12-s + 1.51i·13-s + (0.876 − 0.362i)14-s + (0.269 + 0.269i)15-s − 0.250·16-s + (0.613 − 0.789i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.890 - 0.454i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.890 - 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72813 + 0.414910i\)
\(L(\frac12)\) \(\approx\) \(1.72813 + 0.414910i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.382 - 0.923i)T \)
17 \( 1 + (-2.52 + 3.25i)T \)
good3 \( 1 + (-1.36 + 0.564i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (-1.35 + 3.27i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (4.35 + 1.80i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 5.47iT - 13T^{2} \)
19 \( 1 + (-0.857 - 0.857i)T + 19iT^{2} \)
23 \( 1 + (5.28 + 2.18i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.77 + 4.29i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-6.72 + 2.78i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (8.58 - 3.55i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.0547 + 0.132i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.587 - 0.587i)T - 43iT^{2} \)
47 \( 1 + 5.85iT - 47T^{2} \)
53 \( 1 + (-6.54 - 6.54i)T + 53iT^{2} \)
59 \( 1 + (3.33 - 3.33i)T - 59iT^{2} \)
61 \( 1 + (-4.76 + 11.4i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 8.66T + 67T^{2} \)
71 \( 1 + (-4.75 + 1.96i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-6.17 - 14.9i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-9.34 - 3.87i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-2.40 - 2.40i)T + 83iT^{2} \)
89 \( 1 - 2.24iT - 89T^{2} \)
97 \( 1 + (-1.66 - 4.02i)T + (-68.5 + 68.5i)T^{2} \)
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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

−13.48527967756984319563869752967, −11.93340475843217307734455738669, −10.92717095718889013636888118660, −9.826410352954333617696921078013, −8.299384093439368943093016698727, −7.69476429607915333021088938358, −6.71607394649876030504279754078, −5.21445117041883226244285705797, −3.86255822775029665986653948268, −2.40532095765347409643954785236, 2.26074361973846656271127409516, 3.37017107446711173572010687131, 5.10940936921285466603210446442, 5.75362912733269015170707705360, 7.937539710473713299507249241118, 8.595257446648270767966260020264, 9.794036225432959731058348655474, 10.58944734047855909604616509939, 12.05237874406556887305949261434, 12.58333474375168429035590086074

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