Properties

Label 2-170-17.13-c1-0-3
Degree $2$
Conductor $170$
Sign $0.0245 - 0.999i$
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 + (1.60 + 1.60i)3-s − 4-s + (0.707 + 0.707i)5-s + (−1.60 + 1.60i)6-s + (2.26 − 2.26i)7-s i·8-s + 2.14i·9-s + (−0.707 + 0.707i)10-s + (−1.82 + 1.82i)11-s + (−1.60 − 1.60i)12-s − 5.68·13-s + (2.26 + 2.26i)14-s + 2.26i·15-s + 16-s + (0.604 − 4.07i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.926 + 0.926i)3-s − 0.5·4-s + (0.316 + 0.316i)5-s + (−0.654 + 0.654i)6-s + (0.857 − 0.857i)7-s − 0.353i·8-s + 0.715i·9-s + (−0.223 + 0.223i)10-s + (−0.551 + 0.551i)11-s + (−0.463 − 0.463i)12-s − 1.57·13-s + (0.606 + 0.606i)14-s + 0.585i·15-s + 0.250·16-s + (0.146 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0245 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0245 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.0245 - 0.999i$
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: \((2,\ 170,\ (\ :1/2),\ 0.0245 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06594 + 1.04010i\)
\(L(\frac12)\) \(\approx\) \(1.06594 + 1.04010i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.604 + 4.07i)T \)
good3 \( 1 + (-1.60 - 1.60i)T + 3iT^{2} \)
7 \( 1 + (-2.26 + 2.26i)T - 7iT^{2} \)
11 \( 1 + (1.82 - 1.82i)T - 11iT^{2} \)
13 \( 1 + 5.68T + 13T^{2} \)
19 \( 1 + 4.35iT - 19T^{2} \)
23 \( 1 + (3.41 - 3.41i)T - 23iT^{2} \)
29 \( 1 + (-6.01 - 6.01i)T + 29iT^{2} \)
31 \( 1 + (1.16 + 1.16i)T + 31iT^{2} \)
37 \( 1 + (-3.09 - 3.09i)T + 37iT^{2} \)
41 \( 1 + (-5.53 + 5.53i)T - 41iT^{2} \)
43 \( 1 + 7.20iT - 43T^{2} \)
47 \( 1 + 4.89T + 47T^{2} \)
53 \( 1 - 1.14iT - 53T^{2} \)
59 \( 1 - 8.01iT - 59T^{2} \)
61 \( 1 + (4.51 - 4.51i)T - 61iT^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 + (4.37 + 4.37i)T + 71iT^{2} \)
73 \( 1 + (-3.62 - 3.62i)T + 73iT^{2} \)
79 \( 1 + (1.29 - 1.29i)T - 79iT^{2} \)
83 \( 1 - 12.1iT - 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 + (12.5 + 12.5i)T + 97iT^{2} \)
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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

−13.56871179282570640169069509804, −12.07885349706241551007891917026, −10.59604013823463688419843174026, −9.859326569828399363161885892289, −9.034840822914222927150802003966, −7.72146793648927585253552344953, −7.10962305155156079788852563782, −5.11339095268970229176873326281, −4.38932068598447730550770273819, −2.73645720009660852490064048077, 1.84068066408245972362613001080, 2.73658967997311696735169478097, 4.70079552104114368227734342227, 6.02596376209032578889050019626, 7.988443442900586450041162839848, 8.155535018386360743518581329524, 9.429937576737181083681983957145, 10.50464406893502965531443003600, 11.90664729692119830682302996141, 12.50807741489440927003174769098

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