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Properties

Label 2-845-65.33-c1-0-40
Degree $2$
Conductor $845$
Sign $0.670 + 0.741i$
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  + (0.789 − 1.36i)2-s + (−0.991 + 0.265i)3-s + (−0.247 − 0.428i)4-s + (2.23 + 0.146i)5-s + (−0.419 + 1.56i)6-s + (3.68 − 2.12i)7-s + 2.37·8-s + (−1.68 + 0.973i)9-s + (1.96 − 2.93i)10-s + (0.419 + 1.56i)11-s + (0.358 + 0.358i)12-s − 6.71i·14-s + (−2.25 + 0.447i)15-s + (2.37 − 4.10i)16-s + (−1.60 + 5.98i)17-s + 3.07i·18-s + ⋯
L(s)  = 1  + (0.558 − 0.967i)2-s + (−0.572 + 0.153i)3-s + (−0.123 − 0.214i)4-s + (0.997 + 0.0654i)5-s + (−0.171 + 0.639i)6-s + (1.39 − 0.803i)7-s + 0.840·8-s + (−0.561 + 0.324i)9-s + (0.620 − 0.928i)10-s + (0.126 + 0.472i)11-s + (0.103 + 0.103i)12-s − 1.79i·14-s + (−0.581 + 0.115i)15-s + (0.593 − 1.02i)16-s + (−0.388 + 1.45i)17-s + 0.724i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.670 + 0.741i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.670 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27193 - 1.00897i\)
\(L(\frac12)\) \(\approx\) \(2.27193 - 1.00897i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 - 0.146i)T \)
13 \( 1 \)
good2 \( 1 + (-0.789 + 1.36i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.991 - 0.265i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-3.68 + 2.12i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.419 - 1.56i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.60 - 5.98i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.374 + 0.100i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.600 - 2.24i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.42 + 1.39i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.30 + 2.30i)T + 31iT^{2} \)
37 \( 1 + (1.76 + 1.02i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.20 + 0.323i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.29 - 0.345i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 0.483iT - 47T^{2} \)
53 \( 1 + (7.24 + 7.24i)T + 53iT^{2} \)
59 \( 1 + (0.114 - 0.425i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.64 + 9.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.55 + 6.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.71 + 10.1i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 4.96T + 73T^{2} \)
79 \( 1 - 10.7iT - 79T^{2} \)
83 \( 1 - 11.8iT - 83T^{2} \)
89 \( 1 + (1.50 - 0.404i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.39 + 9.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.58122642776873718762465872380, −9.616910827438961218476037828327, −8.342935156074867951074744820008, −7.58239830733027256827662663107, −6.40627663681629332919048115209, −5.32447364300629607463829240539, −4.68339440158254040544445861200, −3.73695318518444596287437418342, −2.21744946673034502109114757979, −1.51647238840692445057927870626, 1.39568371833415070916172878014, 2.68189279796298010384482048585, 4.61917869577656733223626554066, 5.29351227428448242812917395509, 5.78597634838572998578467996853, 6.58348516497234941462441068371, 7.48964180938354976790491610593, 8.619061690510316093855556357108, 9.162124509377538903473198249841, 10.50249822379784534542613178864

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