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Properties

Label 2-5292-63.5-c1-0-34
Degree $2$
Conductor $5292$
Sign $-0.413 + 0.910i$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
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  + (0.349 + 0.605i)5-s + (−0.229 − 0.132i)11-s + (−1.13 − 0.657i)13-s + (−1.86 − 3.22i)17-s + (0.382 + 0.220i)19-s + (4.29 − 2.48i)23-s + (2.25 − 3.90i)25-s + (0.273 − 0.157i)29-s − 5.60i·31-s + (−0.351 + 0.608i)37-s + (−5.39 + 9.34i)41-s + (3.73 + 6.46i)43-s − 7.00·47-s + (−8.51 + 4.91i)53-s − 0.185i·55-s + ⋯
L(s)  = 1  + (0.156 + 0.270i)5-s + (−0.0692 − 0.0399i)11-s + (−0.315 − 0.182i)13-s + (−0.452 − 0.783i)17-s + (0.0877 + 0.0506i)19-s + (0.896 − 0.517i)23-s + (0.451 − 0.781i)25-s + (0.0507 − 0.0292i)29-s − 1.00i·31-s + (−0.0577 + 0.0999i)37-s + (−0.842 + 1.45i)41-s + (0.569 + 0.985i)43-s − 1.02·47-s + (−1.17 + 0.675i)53-s − 0.0250i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.413 + 0.910i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.413 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.041822543\)
\(L(\frac12)\) \(\approx\) \(1.041822543\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.349 - 0.605i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.229 + 0.132i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.13 + 0.657i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.86 + 3.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.382 - 0.220i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.29 + 2.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.273 + 0.157i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.60iT - 31T^{2} \)
37 \( 1 + (0.351 - 0.608i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.39 - 9.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.73 - 6.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.00T + 47T^{2} \)
53 \( 1 + (8.51 - 4.91i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 5.65iT - 61T^{2} \)
67 \( 1 + 5.94T + 67T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + (-6.66 + 3.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.39T + 79T^{2} \)
83 \( 1 + (3.72 + 6.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.59 + 9.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.18 + 5.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−7.906665083846578723622898991124, −7.28356711025600328763724188333, −6.45196639415625263484398233911, −5.99345174946086117830126352668, −4.73530824121986923746501718667, −4.61767042999517637796646826675, −3.15417288225143997899376162283, −2.74117276800578395593084532796, −1.57246273198988950742622333312, −0.27583486044227342634962154120, 1.23354394312995586812898301245, 2.10233448379506194882402767434, 3.19422529218673002013806984566, 3.90267219962833604303805236255, 4.99996998404339517703572231860, 5.29126345475122965422802326337, 6.39056521502241479377225603947, 6.94240298533649839126532466150, 7.68914093227547498385927946941, 8.504637027257316545923403772838

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