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Properties

Label 2-850-85.22-c1-0-1
Degree $2$
Conductor $850$
Sign $-0.611 - 0.791i$
Analytic cond. $6.78728$
Root an. cond. $2.60524$
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  + (0.382 − 0.923i)2-s + (0.706 + 0.140i)3-s + (−0.707 − 0.707i)4-s + (0.400 − 0.598i)6-s + (−1.70 + 2.55i)7-s + (−0.923 + 0.382i)8-s + (−2.29 − 0.949i)9-s + (−1.25 − 0.838i)11-s + (−0.400 − 0.598i)12-s − 5.77·13-s + (1.70 + 2.55i)14-s + i·16-s + (0.809 + 4.04i)17-s + (−1.75 + 1.75i)18-s + (−1.58 + 0.654i)19-s + ⋯
L(s)  = 1  + (0.270 − 0.653i)2-s + (0.407 + 0.0811i)3-s + (−0.353 − 0.353i)4-s + (0.163 − 0.244i)6-s + (−0.645 + 0.965i)7-s + (−0.326 + 0.135i)8-s + (−0.764 − 0.316i)9-s + (−0.378 − 0.252i)11-s + (−0.115 − 0.172i)12-s − 1.60·13-s + (0.456 + 0.682i)14-s + 0.250i·16-s + (0.196 + 0.980i)17-s + (−0.413 + 0.413i)18-s + (−0.362 + 0.150i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.611 - 0.791i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ -0.611 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0973083 + 0.198081i\)
\(L(\frac12)\) \(\approx\) \(0.0973083 + 0.198081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 \)
17 \( 1 + (-0.809 - 4.04i)T \)
good3 \( 1 + (-0.706 - 0.140i)T + (2.77 + 1.14i)T^{2} \)
7 \( 1 + (1.70 - 2.55i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (1.25 + 0.838i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + 5.77T + 13T^{2} \)
19 \( 1 + (1.58 - 0.654i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.204 + 1.02i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (-0.749 + 3.77i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (3.73 - 2.49i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (1.49 - 7.53i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (0.447 + 2.24i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (2.77 + 6.69i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + (-2.42 - 1.00i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (2.59 - 6.25i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-14.3 + 2.85i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + (11.0 - 11.0i)T - 67iT^{2} \)
71 \( 1 + (-6.97 - 10.4i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (1.30 + 1.94i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (-1.89 + 2.83i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (5.71 - 13.8i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (3.10 + 3.10i)T + 89iT^{2} \)
97 \( 1 + (8.08 + 12.0i)T + (-37.1 + 89.6i)T^{2} \)
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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

−10.26274079427975618212183657402, −9.828617517622865340611695585314, −8.785701559462586167957282709509, −8.347009852533950547961643077165, −6.98953939054678010329650377076, −5.87325841026573072702663714130, −5.24326431723622741524583071016, −3.90619211362283742510446966101, −2.89116104674407369111018638529, −2.20518518908869782233547357869, 0.083966297925478681888963990025, 2.46105042268659839809230124973, 3.40736262928865303555919204234, 4.64076888237650213177880223718, 5.38775107755300907071429681580, 6.59097174877960329937886555231, 7.44633727489165210746655889178, 7.80996723237701475209249850332, 9.099586478221683074601667113425, 9.675451994678620082001791745307

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