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Properties

Label 2-845-169.114-c1-0-51
Degree $2$
Conductor $845$
Sign $-0.452 - 0.891i$
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  + (−0.609 − 0.497i)2-s + (−1.89 − 0.153i)3-s + (−0.276 − 1.35i)4-s + (0.992 − 0.120i)5-s + (1.07 + 1.03i)6-s + (−2.17 − 3.44i)7-s + (−1.23 + 2.35i)8-s + (0.607 + 0.0987i)9-s + (−0.665 − 0.420i)10-s + (−0.876 − 5.39i)11-s + (0.316 + 2.60i)12-s + (3.53 − 0.730i)13-s + (−0.386 + 3.18i)14-s + (−1.89 + 0.0765i)15-s + (−0.614 + 0.261i)16-s + (−0.955 + 0.603i)17-s + ⋯
L(s)  = 1  + (−0.430 − 0.351i)2-s + (−1.09 − 0.0883i)3-s + (−0.138 − 0.676i)4-s + (0.443 − 0.0539i)5-s + (0.440 + 0.423i)6-s + (−0.823 − 1.30i)7-s + (−0.437 + 0.832i)8-s + (0.202 + 0.0329i)9-s + (−0.210 − 0.132i)10-s + (−0.264 − 1.62i)11-s + (0.0913 + 0.752i)12-s + (0.979 − 0.202i)13-s + (−0.103 + 0.850i)14-s + (−0.490 + 0.0197i)15-s + (−0.153 + 0.0654i)16-s + (−0.231 + 0.146i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.452 - 0.891i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.182080 + 0.296754i\)
\(L(\frac12)\) \(\approx\) \(0.182080 + 0.296754i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.992 + 0.120i)T \)
13 \( 1 + (-3.53 + 0.730i)T \)
good2 \( 1 + (0.609 + 0.497i)T + (0.400 + 1.95i)T^{2} \)
3 \( 1 + (1.89 + 0.153i)T + (2.96 + 0.481i)T^{2} \)
7 \( 1 + (2.17 + 3.44i)T + (-3.00 + 6.32i)T^{2} \)
11 \( 1 + (0.876 + 5.39i)T + (-10.4 + 3.48i)T^{2} \)
17 \( 1 + (0.955 - 0.603i)T + (7.28 - 15.3i)T^{2} \)
19 \( 1 + (3.98 + 2.30i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.211 + 0.366i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.21 + 3.93i)T + (-5.80 - 28.4i)T^{2} \)
31 \( 1 + (-1.89 + 7.69i)T + (-27.4 - 14.4i)T^{2} \)
37 \( 1 + (7.68 - 2.22i)T + (31.2 - 19.7i)T^{2} \)
41 \( 1 + (0.192 - 2.38i)T + (-40.4 - 6.57i)T^{2} \)
43 \( 1 + (2.18 - 7.55i)T + (-36.3 - 22.9i)T^{2} \)
47 \( 1 + (3.70 - 4.18i)T + (-5.66 - 46.6i)T^{2} \)
53 \( 1 + (-9.37 - 4.92i)T + (30.1 + 43.6i)T^{2} \)
59 \( 1 + (0.348 - 0.818i)T + (-40.8 - 42.5i)T^{2} \)
61 \( 1 + (0.0642 - 1.59i)T + (-60.8 - 4.90i)T^{2} \)
67 \( 1 + (8.26 + 1.68i)T + (61.6 + 26.2i)T^{2} \)
71 \( 1 + (-7.43 + 3.53i)T + (44.9 - 54.9i)T^{2} \)
73 \( 1 + (0.700 + 0.265i)T + (54.6 + 48.4i)T^{2} \)
79 \( 1 + (-10.0 - 8.92i)T + (9.52 + 78.4i)T^{2} \)
83 \( 1 + (9.83 - 6.78i)T + (29.4 - 77.6i)T^{2} \)
89 \( 1 + (2.60 - 1.50i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.85 + 9.11i)T + (-26.9 - 93.1i)T^{2} \)
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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

−9.947229909876922150474832645574, −8.906161517010588999519392042657, −8.141019923855045466483113672757, −6.44917906638520959343897805927, −6.30608088635432731159390386319, −5.43913530025420880051297011881, −4.23229159952115705956818055749, −2.90853879640436593584015960543, −1.05850994578810106130211669096, −0.26998336407191422138667734980, 2.12349945183547799797549552075, 3.42722552179146112891281395760, 4.77844307792021831168179278132, 5.66668196264791610255463538329, 6.57501247751346552461904261453, 6.99700457677024284005979427120, 8.573193150145198684831603842288, 8.850559799268643462303310681862, 9.967780834179650000601251181922, 10.53193746959471369681150963792

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