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Properties

Label 2-170-5.4-c1-0-3
Degree $2$
Conductor $170$
Sign $0.807 - 0.590i$
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 − 0.484i·3-s − 4-s + (1.80 − 1.32i)5-s + 0.484·6-s + 2.64i·7-s − i·8-s + 2.76·9-s + (1.32 + 1.80i)10-s + 2·11-s + 0.484i·12-s − 0.484i·13-s − 2.64·14-s + (−0.640 − 0.875i)15-s + 16-s + i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.279i·3-s − 0.5·4-s + (0.807 − 0.590i)5-s + 0.197·6-s + 0.997i·7-s − 0.353i·8-s + 0.921·9-s + (0.417 + 0.570i)10-s + 0.603·11-s + 0.139i·12-s − 0.134i·13-s − 0.705·14-s + (−0.165 − 0.225i)15-s + 0.250·16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.807 - 0.590i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.807 - 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21053 + 0.395433i\)
\(L(\frac12)\) \(\approx\) \(1.21053 + 0.395433i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 + 1.32i)T \)
17 \( 1 - iT \)
good3 \( 1 + 0.484iT - 3T^{2} \)
7 \( 1 - 2.64iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 0.484iT - 13T^{2} \)
19 \( 1 + 5.76T + 19T^{2} \)
23 \( 1 - 1.35iT - 23T^{2} \)
29 \( 1 + 8.15T + 29T^{2} \)
31 \( 1 + 2.09T + 31T^{2} \)
37 \( 1 + 11.1iT - 37T^{2} \)
41 \( 1 + 0.249T + 41T^{2} \)
43 \( 1 + 1.03iT - 43T^{2} \)
47 \( 1 - 6.01iT - 47T^{2} \)
53 \( 1 - 7.70iT - 53T^{2} \)
59 \( 1 + 8.73T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + 4.96iT - 67T^{2} \)
71 \( 1 - 8.34T + 71T^{2} \)
73 \( 1 + 0.484iT - 73T^{2} \)
79 \( 1 + 9.85T + 79T^{2} \)
83 \( 1 - 17.5iT - 83T^{2} \)
89 \( 1 - 8.73T + 89T^{2} \)
97 \( 1 - 6.73iT - 97T^{2} \)
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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

−12.76629568955812884546618601968, −12.41866790205771468703844216335, −10.76680111784602260745190539806, −9.425717503589357470578108740550, −8.931717149364140808607622622846, −7.65115084853144929419777824949, −6.37357647096275061077695188600, −5.57962699285094485382833708805, −4.23346072278715261336469673194, −1.89720428254307445804734850852, 1.76264529331478090908620929120, 3.57712463063428713047093127538, 4.66520595412946840901532946177, 6.35477118651504207866264344498, 7.36136494111513633907947716182, 8.983604336757551229256098243215, 9.979952050305789107577569733864, 10.52047979950124665343532625219, 11.45178978154267891317040345832, 12.85101135375405918660384478273

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