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Properties

Label 2-170-85.49-c1-0-7
Degree $2$
Conductor $170$
Sign $0.864 + 0.503i$
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 + (2.33 − 0.969i)3-s − 1.00i·4-s + (−1.78 − 1.34i)5-s + (−0.969 + 2.33i)6-s + (1.26 − 3.06i)7-s + (0.707 + 0.707i)8-s + (2.41 − 2.41i)9-s + (2.21 − 0.309i)10-s + (−0.949 + 2.29i)11-s + (−0.969 − 2.33i)12-s + 0.777·13-s + (1.26 + 3.06i)14-s + (−5.48 − 1.42i)15-s − 1.00·16-s + (−4.04 + 0.798i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (1.35 − 0.559i)3-s − 0.500i·4-s + (−0.798 − 0.602i)5-s + (−0.395 + 0.955i)6-s + (0.479 − 1.15i)7-s + (0.250 + 0.250i)8-s + (0.804 − 0.804i)9-s + (0.700 − 0.0979i)10-s + (−0.286 + 0.691i)11-s + (−0.279 − 0.675i)12-s + 0.215·13-s + (0.338 + 0.817i)14-s + (−1.41 − 0.367i)15-s − 0.250·16-s + (−0.981 + 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.864 + 0.503i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.864 + 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18744 - 0.320644i\)
\(L(\frac12)\) \(\approx\) \(1.18744 - 0.320644i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.78 + 1.34i)T \)
17 \( 1 + (4.04 - 0.798i)T \)
good3 \( 1 + (-2.33 + 0.969i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (-1.26 + 3.06i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.949 - 2.29i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 0.777T + 13T^{2} \)
19 \( 1 + (-4.13 - 4.13i)T + 19iT^{2} \)
23 \( 1 + (-7.45 - 3.08i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-0.319 + 0.132i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.719 + 1.73i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (4.26 - 1.76i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.17 - 0.900i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (8.03 + 8.03i)T + 43iT^{2} \)
47 \( 1 + 9.25T + 47T^{2} \)
53 \( 1 + (-2.98 + 2.98i)T - 53iT^{2} \)
59 \( 1 + (9.06 - 9.06i)T - 59iT^{2} \)
61 \( 1 + (-12.7 - 5.29i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 0.862iT - 67T^{2} \)
71 \( 1 + (3.13 + 7.57i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.75 + 9.05i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.33 + 8.04i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-2.56 + 2.56i)T - 83iT^{2} \)
89 \( 1 + 3.67iT - 89T^{2} \)
97 \( 1 + (-1.05 - 2.53i)T + (-68.5 + 68.5i)T^{2} \)
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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

−13.11435322907869146149271372017, −11.71398130238526752435654543961, −10.49726226883412059797283098785, −9.255709740864374317839161376924, −8.404362721486378877131143141626, −7.58067804227413407845214716363, −7.07392215549510738109647965067, −4.90132599448890860681851807811, −3.58086371555703406492834771095, −1.49228472378177988778540105207, 2.58432760344629849439845300110, 3.30271765480785716061690047039, 4.84217624369133522827559534116, 6.94011632235486915765007867363, 8.279013165409103909480620129630, 8.680516555889157201867699415194, 9.602001023152909072093681950260, 11.00369523814161570725013789956, 11.47925119610958882996282176569, 12.84867304619788600280987560155

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