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Properties

Label 2-85-17.4-c1-0-4
Degree $2$
Conductor $85$
Sign $0.722 + 0.691i$
Analytic cond. $0.678728$
Root an. cond. $0.823849$
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  − 0.783i·2-s + (0.385 − 0.385i)3-s + 1.38·4-s + (−0.707 + 0.707i)5-s + (−0.301 − 0.301i)6-s + (−0.840 − 0.840i)7-s − 2.65i·8-s + 2.70i·9-s + (0.554 + 0.554i)10-s + (−1.80 − 1.80i)11-s + (0.534 − 0.534i)12-s − 0.368·13-s + (−0.658 + 0.658i)14-s + 0.544i·15-s + 0.693·16-s + (−2.46 + 3.30i)17-s + ⋯
L(s)  = 1  − 0.554i·2-s + (0.222 − 0.222i)3-s + 0.693·4-s + (−0.316 + 0.316i)5-s + (−0.123 − 0.123i)6-s + (−0.317 − 0.317i)7-s − 0.937i·8-s + 0.901i·9-s + (0.175 + 0.175i)10-s + (−0.544 − 0.544i)11-s + (0.154 − 0.154i)12-s − 0.102·13-s + (−0.175 + 0.175i)14-s + 0.140i·15-s + 0.173·16-s + (−0.597 + 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00714 - 0.404251i\)
\(L(\frac12)\) \(\approx\) \(1.00714 - 0.404251i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (2.46 - 3.30i)T \)
good2 \( 1 + 0.783iT - 2T^{2} \)
3 \( 1 + (-0.385 + 0.385i)T - 3iT^{2} \)
7 \( 1 + (0.840 + 0.840i)T + 7iT^{2} \)
11 \( 1 + (1.80 + 1.80i)T + 11iT^{2} \)
13 \( 1 + 0.368T + 13T^{2} \)
19 \( 1 - 6.61iT - 19T^{2} \)
23 \( 1 + (2.73 + 2.73i)T + 23iT^{2} \)
29 \( 1 + (1.63 - 1.63i)T - 29iT^{2} \)
31 \( 1 + (-4.68 + 4.68i)T - 31iT^{2} \)
37 \( 1 + (-2.24 + 2.24i)T - 37iT^{2} \)
41 \( 1 + (5.16 + 5.16i)T + 41iT^{2} \)
43 \( 1 + 6.82iT - 43T^{2} \)
47 \( 1 - 7.80T + 47T^{2} \)
53 \( 1 - 8.01iT - 53T^{2} \)
59 \( 1 + 5.22iT - 59T^{2} \)
61 \( 1 + (-5.74 - 5.74i)T + 61iT^{2} \)
67 \( 1 + 7.94T + 67T^{2} \)
71 \( 1 + (-8.40 + 8.40i)T - 71iT^{2} \)
73 \( 1 + (10.4 - 10.4i)T - 73iT^{2} \)
79 \( 1 + (-0.575 - 0.575i)T + 79iT^{2} \)
83 \( 1 - 3.99iT - 83T^{2} \)
89 \( 1 - 9.14T + 89T^{2} \)
97 \( 1 + (4.99 - 4.99i)T - 97iT^{2} \)
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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

−13.95653431690253372012910852823, −12.94746818154322777360271289218, −11.94203746056054755701305760104, −10.71814071240957932579666241135, −10.24850616838637536926903791885, −8.314174624295601126789626816942, −7.31987578503170238547140125632, −5.98115673283250579907771985633, −3.81048101285519635290577436302, −2.26741207881119474684057189381, 2.82052032323703283600060930000, 4.82488185927175170488320408520, 6.36604859368559251388368254162, 7.39704706972795222354855796452, 8.713547089346197654991642266000, 9.810928896040372664563587986831, 11.32005489151324257170062682026, 12.13909247776387009204830596274, 13.38250916672018972629140759132, 14.76968241555613600492334305049

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