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Properties

Label 2-845-65.57-c1-0-57
Degree $2$
Conductor $845$
Sign $-0.567 + 0.823i$
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  + 0.274·2-s + (1.67 − 1.67i)3-s − 1.92·4-s + (1.69 − 1.45i)5-s + (0.459 − 0.459i)6-s + 0.386i·7-s − 1.07·8-s − 2.58i·9-s + (0.466 − 0.399i)10-s + (−3.08 − 3.08i)11-s + (−3.21 + 3.21i)12-s + 0.106i·14-s + (0.409 − 5.26i)15-s + 3.55·16-s + (1.39 − 1.39i)17-s − 0.710i·18-s + ⋯
L(s)  = 1  + 0.194·2-s + (0.964 − 0.964i)3-s − 0.962·4-s + (0.759 − 0.650i)5-s + (0.187 − 0.187i)6-s + 0.145i·7-s − 0.381·8-s − 0.861i·9-s + (0.147 − 0.126i)10-s + (−0.929 − 0.929i)11-s + (−0.928 + 0.928i)12-s + 0.0283i·14-s + (0.105 − 1.36i)15-s + 0.888·16-s + (0.338 − 0.338i)17-s − 0.167i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.567 + 0.823i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.567 + 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.844163 - 1.60683i\)
\(L(\frac12)\) \(\approx\) \(0.844163 - 1.60683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.69 + 1.45i)T \)
13 \( 1 \)
good2 \( 1 - 0.274T + 2T^{2} \)
3 \( 1 + (-1.67 + 1.67i)T - 3iT^{2} \)
7 \( 1 - 0.386iT - 7T^{2} \)
11 \( 1 + (3.08 + 3.08i)T + 11iT^{2} \)
17 \( 1 + (-1.39 + 1.39i)T - 17iT^{2} \)
19 \( 1 + (3.54 + 3.54i)T + 19iT^{2} \)
23 \( 1 + (-0.235 - 0.235i)T + 23iT^{2} \)
29 \( 1 + 8.16iT - 29T^{2} \)
31 \( 1 + (2.54 - 2.54i)T - 31iT^{2} \)
37 \( 1 - 4.82iT - 37T^{2} \)
41 \( 1 + (3.29 - 3.29i)T - 41iT^{2} \)
43 \( 1 + (-4.82 - 4.82i)T + 43iT^{2} \)
47 \( 1 + 9.83iT - 47T^{2} \)
53 \( 1 + (7.17 - 7.17i)T - 53iT^{2} \)
59 \( 1 + (-1.71 + 1.71i)T - 59iT^{2} \)
61 \( 1 - 10.6T + 61T^{2} \)
67 \( 1 + 6.37T + 67T^{2} \)
71 \( 1 + (3.07 - 3.07i)T - 71iT^{2} \)
73 \( 1 - 6.08T + 73T^{2} \)
79 \( 1 + 3.34iT - 79T^{2} \)
83 \( 1 + 5.18iT - 83T^{2} \)
89 \( 1 + (-3.53 + 3.53i)T - 89iT^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−9.639320064665692319356438740563, −8.796304882518994457065620036662, −8.403065703812152763183788167073, −7.62451891544483466292984716016, −6.33236756611319303492230009621, −5.46382634174967644493163048699, −4.58155517073847105943816916369, −3.17314070371739596669465822431, −2.23931131370059297619630127484, −0.75564977346626196480193070224, 2.11665266770997857182875952248, 3.26167525571816232988289820609, 4.04826169490676460524484980448, 5.01283924376315146525496515147, 5.86271689037025876460292851223, 7.20985249519976377703675907321, 8.173420783671260125091752948303, 9.002767537172791458395590884597, 9.650811013979799466741085258250, 10.31220673746628285906941047127

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