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Properties

Label 2-85-17.16-c3-0-2
Degree $2$
Conductor $85$
Sign $-0.970 - 0.242i$
Analytic cond. $5.01516$
Root an. cond. $2.23945$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 8i·3-s − 7·4-s + 5i·5-s + 8i·6-s − 14i·7-s − 15·8-s − 37·9-s + 5i·10-s + 20i·11-s − 56i·12-s − 58·13-s − 14i·14-s − 40·15-s + 41·16-s + (−17 + 68i)17-s + ⋯
L(s)  = 1  + 0.353·2-s + 1.53i·3-s − 0.875·4-s + 0.447i·5-s + 0.544i·6-s − 0.755i·7-s − 0.662·8-s − 1.37·9-s + 0.158i·10-s + 0.548i·11-s − 1.34i·12-s − 1.23·13-s − 0.267i·14-s − 0.688·15-s + 0.640·16-s + (−0.242 + 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.970 - 0.242i$
Analytic conductor: \(5.01516\)
Root analytic conductor: \(2.23945\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :3/2),\ -0.970 - 0.242i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.119594 + 0.971482i\)
\(L(\frac12)\) \(\approx\) \(0.119594 + 0.971482i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 5iT \)
17 \( 1 + (17 - 68i)T \)
good2 \( 1 - T + 8T^{2} \)
3 \( 1 - 8iT - 27T^{2} \)
7 \( 1 + 14iT - 343T^{2} \)
11 \( 1 - 20iT - 1.33e3T^{2} \)
13 \( 1 + 58T + 2.19e3T^{2} \)
19 \( 1 - 80T + 6.85e3T^{2} \)
23 \( 1 - 118iT - 1.21e4T^{2} \)
29 \( 1 - 126iT - 2.43e4T^{2} \)
31 \( 1 - 70iT - 2.97e4T^{2} \)
37 \( 1 + 134iT - 5.06e4T^{2} \)
41 \( 1 + 100iT - 6.89e4T^{2} \)
43 \( 1 - 272T + 7.95e4T^{2} \)
47 \( 1 + 464T + 1.03e5T^{2} \)
53 \( 1 - 642T + 1.48e5T^{2} \)
59 \( 1 + 180T + 2.05e5T^{2} \)
61 \( 1 - 110iT - 2.26e5T^{2} \)
67 \( 1 + 924T + 3.00e5T^{2} \)
71 \( 1 - 90iT - 3.57e5T^{2} \)
73 \( 1 - 828iT - 3.89e5T^{2} \)
79 \( 1 + 1.33e3iT - 4.93e5T^{2} \)
83 \( 1 - 552T + 5.71e5T^{2} \)
89 \( 1 - 1.49e3T + 7.04e5T^{2} \)
97 \( 1 - 1.37e3iT - 9.12e5T^{2} \)
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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

−14.53212581955990699818252781324, −13.42173023617267774394462262568, −12.07821564623242137251481014525, −10.62655220097931137227209577562, −9.919390906428201803375419174620, −9.082915190059453143991000377284, −7.42401284624272390477745900988, −5.43357381620297151217858496435, −4.42162856375152761553122566849, −3.43175653740013403079307026425, 0.54316769654052021895155318448, 2.63325124691442724915180040174, 4.90144013001537491163118703975, 6.05756273347477248130378560198, 7.52356065874312506569484692299, 8.580253255598576001975807556965, 9.619029205181562761772609619199, 11.80037785983210153958294168533, 12.28376469979931862577558217965, 13.27573373967748519700682373569

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