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Properties

Label 2-170-17.13-c1-0-2
Degree $2$
Conductor $170$
Sign $0.995 - 0.0929i$
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  − i·2-s + (1.70 + 1.70i)3-s − 4-s + (0.707 + 0.707i)5-s + (1.70 − 1.70i)6-s + (−0.414 + 0.414i)7-s + i·8-s + 2.82i·9-s + (0.707 − 0.707i)10-s + (1 − i)11-s + (−1.70 − 1.70i)12-s + 13-s + (0.414 + 0.414i)14-s + 2.41i·15-s + 16-s + (−4.12 − 0.121i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.985 + 0.985i)3-s − 0.5·4-s + (0.316 + 0.316i)5-s + (0.696 − 0.696i)6-s + (−0.156 + 0.156i)7-s + 0.353i·8-s + 0.942i·9-s + (0.223 − 0.223i)10-s + (0.301 − 0.301i)11-s + (−0.492 − 0.492i)12-s + 0.277·13-s + (0.110 + 0.110i)14-s + 0.623i·15-s + 0.250·16-s + (−0.999 − 0.0294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.995 - 0.0929i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.995 - 0.0929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47694 + 0.0687724i\)
\(L(\frac12)\) \(\approx\) \(1.47694 + 0.0687724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (4.12 + 0.121i)T \)
good3 \( 1 + (-1.70 - 1.70i)T + 3iT^{2} \)
7 \( 1 + (0.414 - 0.414i)T - 7iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 - T + 13T^{2} \)
19 \( 1 + 2.41iT - 19T^{2} \)
23 \( 1 + (-2.24 + 2.24i)T - 23iT^{2} \)
29 \( 1 + (6.94 + 6.94i)T + 29iT^{2} \)
31 \( 1 + (3.70 + 3.70i)T + 31iT^{2} \)
37 \( 1 + (-1.58 - 1.58i)T + 37iT^{2} \)
41 \( 1 + (6.65 - 6.65i)T - 41iT^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 - 3.24T + 47T^{2} \)
53 \( 1 + 3.48iT - 53T^{2} \)
59 \( 1 + 10.8iT - 59T^{2} \)
61 \( 1 + (5.77 - 5.77i)T - 61iT^{2} \)
67 \( 1 - 8.82T + 67T^{2} \)
71 \( 1 + (8.29 + 8.29i)T + 71iT^{2} \)
73 \( 1 + (-5.53 - 5.53i)T + 73iT^{2} \)
79 \( 1 + (-8.24 + 8.24i)T - 79iT^{2} \)
83 \( 1 - 9.89iT - 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 + (-10.1 - 10.1i)T + 97iT^{2} \)
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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

−13.03270571690765864980962757484, −11.48556577297951408441499229173, −10.75976768697985784411792928055, −9.573415864274829646491736761962, −9.172365663510931368354358925966, −8.050707548346166946578814107921, −6.32062236175296613738540838168, −4.66807578391039379930519681619, −3.56307510512022131272489367799, −2.42330288948026828083426466824, 1.81235193467251682123806649195, 3.66162692050211556891651298087, 5.36873415436402762503032600660, 6.78339980624117990803504284231, 7.43053067822004441243576700880, 8.682080868302953052200804921101, 9.152155523630088363683499822339, 10.63307754013101297553725857132, 12.23390483141118039442562130552, 13.08933133471289730069534100529

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