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Properties

Label 2-170-17.4-c1-0-3
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: $\chi_{170} (21, \cdot )$
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

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

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