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Properties

Label 2-845-65.49-c1-0-32
Degree $2$
Conductor $845$
Sign $-0.0623 - 0.998i$
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  + (0.607 + 1.05i)2-s + (−1.13 + 0.655i)3-s + (0.262 − 0.455i)4-s + (0.311 + 2.21i)5-s + (−1.37 − 0.796i)6-s + (1.45 − 2.51i)7-s + 3.06·8-s + (−0.640 + 1.10i)9-s + (−2.13 + 1.67i)10-s + (0.185 − 0.107i)11-s + 0.688i·12-s + 3.52·14-s + (−1.80 − 2.31i)15-s + (1.33 + 2.31i)16-s + (5.56 + 3.21i)17-s − 1.55·18-s + ⋯
L(s)  = 1  + (0.429 + 0.743i)2-s + (−0.655 + 0.378i)3-s + (0.131 − 0.227i)4-s + (0.139 + 0.990i)5-s + (−0.562 − 0.324i)6-s + (0.548 − 0.950i)7-s + 1.08·8-s + (−0.213 + 0.369i)9-s + (−0.676 + 0.528i)10-s + (0.0559 − 0.0323i)11-s + 0.198i·12-s + 0.942·14-s + (−0.466 − 0.596i)15-s + (0.334 + 0.578i)16-s + (1.35 + 0.779i)17-s − 0.366·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0623 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0623 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.0623 - 0.998i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.0623 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31811 + 1.40297i\)
\(L(\frac12)\) \(\approx\) \(1.31811 + 1.40297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.311 - 2.21i)T \)
13 \( 1 \)
good2 \( 1 + (-0.607 - 1.05i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.13 - 0.655i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.45 + 2.51i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.185 + 0.107i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-5.56 - 3.21i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.91 + 1.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.06 + 2.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.35 - 7.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.59iT - 31T^{2} \)
37 \( 1 + (-1.14 - 1.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.64 - 1.52i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.50 + 3.18i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.09T + 47T^{2} \)
53 \( 1 - 6.23iT - 53T^{2} \)
59 \( 1 + (8.02 + 4.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.140 + 0.243i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.88 - 6.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.26 + 3.04i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 9.52T + 83T^{2} \)
89 \( 1 + (-4.86 + 2.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.02 + 15.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.58882269832958403318350339909, −10.06719754215831766121638911823, −8.437598243398214872472874880892, −7.53182359790330395572074455594, −6.84189116418053492841649149733, −6.07746207862146574932133208554, −5.17784900272502231594987004866, −4.47347015232459354918329310815, −3.17668484429524223076857591105, −1.47698453695986166873603540821, 1.02781284350167616449447770322, 2.20740439666095851770032035167, 3.42393007125275699713273423216, 4.68480350062719586015400110699, 5.39385358784405916576051060853, 6.23077044712588594524816903151, 7.53326900067099192871938551296, 8.257825172546624501927798839980, 9.204241377698220101082038363391, 10.06288733140634050236762695058

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