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Properties

Label 2-845-5.4-c1-0-51
Degree $2$
Conductor $845$
Sign $0.204 + 0.978i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  − 0.456i·2-s + i·3-s + 1.79·4-s + (−0.456 − 2.18i)5-s + 0.456·6-s − 1.73i·7-s − 1.73i·8-s + 2·9-s + (−0.999 + 0.208i)10-s − 2.64·11-s + 1.79i·12-s − 0.791·14-s + (2.18 − 0.456i)15-s + 2.79·16-s − 4.58i·17-s − 0.913i·18-s + ⋯
L(s)  = 1  − 0.323i·2-s + 0.577i·3-s + 0.895·4-s + (−0.204 − 0.978i)5-s + 0.186·6-s − 0.654i·7-s − 0.612i·8-s + 0.666·9-s + (−0.316 + 0.0660i)10-s − 0.797·11-s + 0.517i·12-s − 0.211·14-s + (0.565 − 0.117i)15-s + 0.697·16-s − 1.11i·17-s − 0.215i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40020 - 1.13813i\)
\(L(\frac12)\) \(\approx\) \(1.40020 - 1.13813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.456 + 2.18i)T \)
13 \( 1 \)
good2 \( 1 + 0.456iT - 2T^{2} \)
3 \( 1 - iT - 3T^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + 2.64T + 11T^{2} \)
17 \( 1 + 4.58iT - 17T^{2} \)
19 \( 1 + 1.73T + 19T^{2} \)
23 \( 1 + 4.58iT - 23T^{2} \)
29 \( 1 + 4.58T + 29T^{2} \)
31 \( 1 - 6.20T + 31T^{2} \)
37 \( 1 - 7.93iT - 37T^{2} \)
41 \( 1 + 2.64T + 41T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 + 1.82iT - 47T^{2} \)
53 \( 1 - 7.58iT - 53T^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 - 1.00iT - 67T^{2} \)
71 \( 1 - 7.02T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 6.01iT - 83T^{2} \)
89 \( 1 - 9.57T + 89T^{2} \)
97 \( 1 - 11.4iT - 97T^{2} \)
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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

−10.22018131844077347397055224048, −9.380743873630575181318871635409, −8.309647964266327307947642663870, −7.43776167962825481759029302938, −6.69925734714272603908445846248, −5.36123587215514942260709120920, −4.54409487211441825649869448520, −3.64029158919873502371472766363, −2.32547281765065705870897333403, −0.879848461701880786751774590710, 1.83604315401163777778163997195, 2.63875161272814276193137891620, 3.84760204373489446658633377495, 5.44917616550175838112927065786, 6.24807554457554457474090755857, 6.91884732017736151672712684575, 7.73968700310539788197428313505, 8.230024990396930936436046946174, 9.700989360576677484375068353218, 10.50912632847757358281976561600

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