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Properties

Label 2-845-65.33-c1-0-21
Degree $2$
Conductor $845$
Sign $-0.378 - 0.925i$
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  + (−1.12 + 1.94i)2-s + (−1.91 + 0.514i)3-s + (−1.53 − 2.65i)4-s + (2.22 − 0.247i)5-s + (1.15 − 4.31i)6-s + (−1.10 + 0.638i)7-s + 2.39·8-s + (0.820 − 0.473i)9-s + (−2.01 + 4.61i)10-s + (−1.41 − 5.27i)11-s + (4.30 + 4.30i)12-s − 2.87i·14-s + (−4.13 + 1.61i)15-s + (0.365 − 0.633i)16-s + (−0.833 + 3.11i)17-s + 2.13i·18-s + ⋯
L(s)  = 1  + (−0.795 + 1.37i)2-s + (−1.10 + 0.296i)3-s + (−0.766 − 1.32i)4-s + (0.993 − 0.110i)5-s + (0.472 − 1.76i)6-s + (−0.418 + 0.241i)7-s + 0.848·8-s + (0.273 − 0.157i)9-s + (−0.638 + 1.45i)10-s + (−0.426 − 1.59i)11-s + (1.24 + 1.24i)12-s − 0.768i·14-s + (−1.06 + 0.417i)15-s + (0.0913 − 0.158i)16-s + (−0.202 + 0.754i)17-s + 0.502i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.378 - 0.925i$
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: \((2,\ 845,\ (\ :1/2),\ -0.378 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.356096 + 0.530213i\)
\(L(\frac12)\) \(\approx\) \(0.356096 + 0.530213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.22 + 0.247i)T \)
13 \( 1 \)
good2 \( 1 + (1.12 - 1.94i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.91 - 0.514i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.10 - 0.638i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.41 + 5.27i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.833 - 3.11i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.17 - 0.315i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.0428 + 0.160i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-8.41 - 4.85i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.233 + 0.233i)T + 31iT^{2} \)
37 \( 1 + (1.14 + 0.660i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.483 + 0.129i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.43 - 1.72i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 3.20iT - 47T^{2} \)
53 \( 1 + (-4.49 - 4.49i)T + 53iT^{2} \)
59 \( 1 + (0.000595 - 0.00222i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.695 + 1.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.03 + 5.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.14 + 11.7i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 7.34T + 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + 2.65iT - 83T^{2} \)
89 \( 1 + (-6.96 + 1.86i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.09 - 3.62i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.46846867327671345124894787694, −9.404292031787358542479064268237, −8.734033805202795154819412135573, −7.999504743927031934099709275104, −6.67054626355640750157858527017, −6.08599165893383704243377462047, −5.67967155588078831917187257874, −4.84930240495011226664262553238, −2.96378508706625432891826471324, −0.827955740471077355126066712508, 0.70840668876880470192964928534, 2.00457588243729505543616795913, 2.90128488376296081358098530515, 4.49147638408884843542787080880, 5.48559994610782945430580687425, 6.53568616304963332822771565844, 7.26991280438859526621626574587, 8.623249805188768209556024538358, 9.587249271059987470135952107484, 10.06607018686048206271395619531

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