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Properties

Label 2-845-65.28-c1-0-39
Degree $2$
Conductor $845$
Sign $0.499 - 0.866i$
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  + (−0.885 + 0.511i)2-s + (−0.721 + 2.69i)3-s + (−0.477 + 0.826i)4-s + (1.45 − 1.69i)5-s + (−0.737 − 2.75i)6-s + (0.481 − 0.834i)7-s − 3.02i·8-s + (−4.12 − 2.38i)9-s + (−0.423 + 2.24i)10-s + (0.430 − 1.60i)11-s + (−1.88 − 1.88i)12-s + 0.985i·14-s + (3.51 + 5.14i)15-s + (0.590 + 1.02i)16-s + (7.00 − 1.87i)17-s + 4.87·18-s + ⋯
L(s)  = 1  + (−0.626 + 0.361i)2-s + (−0.416 + 1.55i)3-s + (−0.238 + 0.413i)4-s + (0.651 − 0.758i)5-s + (−0.301 − 1.12i)6-s + (0.182 − 0.315i)7-s − 1.06i·8-s + (−1.37 − 0.794i)9-s + (−0.133 + 0.710i)10-s + (0.129 − 0.484i)11-s + (−0.542 − 0.542i)12-s + 0.263i·14-s + (0.907 + 1.32i)15-s + (0.147 + 0.255i)16-s + (1.69 − 0.455i)17-s + 1.14·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.499 - 0.866i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.499 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.885770 + 0.511672i\)
\(L(\frac12)\) \(\approx\) \(0.885770 + 0.511672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.45 + 1.69i)T \)
13 \( 1 \)
good2 \( 1 + (0.885 - 0.511i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.721 - 2.69i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.481 + 0.834i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.430 + 1.60i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-7.00 + 1.87i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.64 + 0.707i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.72 - 0.997i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.253 + 0.146i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.125 + 0.125i)T - 31iT^{2} \)
37 \( 1 + (2.04 + 3.53i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.69 + 1.79i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.05 + 7.67i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 7.84T + 47T^{2} \)
53 \( 1 + (1.99 + 1.99i)T + 53iT^{2} \)
59 \( 1 + (1.30 + 4.87i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.04 - 1.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.32 + 3.64i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.37 - 12.6i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 3.22iT - 73T^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 - 8.56T + 83T^{2} \)
89 \( 1 + (0.500 + 0.134i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.50 + 3.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.02631289238906676012390090906, −9.503655844408591204577677543410, −8.904527544627447795053493277293, −8.072509064264542113481938577206, −7.00509114769537421685174642374, −5.61369022554950875773990913031, −5.12841354302598861045524134927, −4.05201414963801530814618413955, −3.23980569404971541847241050644, −0.840239350458734116899931495616, 1.14389057678432726791314157561, 1.89512714422108636181565133897, 3.02897189988179498819880876100, 5.15505074447522236999876374774, 5.81710869885207503797804591678, 6.63269200968377286678691061581, 7.52397775851728811484150382439, 8.239074990246099924466561499933, 9.336305654792670622075260532991, 10.10540690398641762241766359721

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