Properties

Label 2-2e5-8.5-c7-0-0
Degree $2$
Conductor $32$
Sign $-0.137 - 0.990i$
Analytic cond. $9.99632$
Root an. cond. $3.16169$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 21.9i·3-s + 184. i·5-s − 1.05e3·7-s + 1.70e3·9-s + 4.32e3i·11-s + 1.12e4i·13-s + 4.05e3·15-s − 2.17e4·17-s + 4.54e4i·19-s + 2.30e4i·21-s − 4.41e3·23-s + 4.39e4·25-s − 8.54e4i·27-s − 2.36e4i·29-s − 7.29e4·31-s + ⋯
L(s)  = 1  − 0.469i·3-s + 0.661i·5-s − 1.15·7-s + 0.779·9-s + 0.979i·11-s + 1.42i·13-s + 0.310·15-s − 1.07·17-s + 1.52i·19-s + 0.543i·21-s − 0.0756·23-s + 0.562·25-s − 0.835i·27-s − 0.180i·29-s − 0.439·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.137 - 0.990i$
Analytic conductor: \(9.99632\)
Root analytic conductor: \(3.16169\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :7/2),\ -0.137 - 0.990i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.727854 + 0.835600i\)
\(L(\frac12)\) \(\approx\) \(0.727854 + 0.835600i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 21.9iT - 2.18e3T^{2} \)
5 \( 1 - 184. iT - 7.81e4T^{2} \)
7 \( 1 + 1.05e3T + 8.23e5T^{2} \)
11 \( 1 - 4.32e3iT - 1.94e7T^{2} \)
13 \( 1 - 1.12e4iT - 6.27e7T^{2} \)
17 \( 1 + 2.17e4T + 4.10e8T^{2} \)
19 \( 1 - 4.54e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.41e3T + 3.40e9T^{2} \)
29 \( 1 + 2.36e4iT - 1.72e10T^{2} \)
31 \( 1 + 7.29e4T + 2.75e10T^{2} \)
37 \( 1 + 4.83e5iT - 9.49e10T^{2} \)
41 \( 1 + 4.11e5T + 1.94e11T^{2} \)
43 \( 1 + 9.61e4iT - 2.71e11T^{2} \)
47 \( 1 - 1.56e5T + 5.06e11T^{2} \)
53 \( 1 - 6.86e5iT - 1.17e12T^{2} \)
59 \( 1 - 1.79e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.36e6iT - 3.14e12T^{2} \)
67 \( 1 + 1.08e6iT - 6.06e12T^{2} \)
71 \( 1 - 5.60e6T + 9.09e12T^{2} \)
73 \( 1 - 2.16e4T + 1.10e13T^{2} \)
79 \( 1 + 2.34e6T + 1.92e13T^{2} \)
83 \( 1 + 8.82e5iT - 2.71e13T^{2} \)
89 \( 1 + 1.34e6T + 4.42e13T^{2} \)
97 \( 1 - 7.32e6T + 8.07e13T^{2} \)
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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

−15.61601200075383968768194692901, −14.32227447456084385506275002719, −13.05357997620429506881090551382, −12.09809966957710376484747233310, −10.41954573956791887129760552716, −9.310780730292977823859471632986, −7.23449504136212369919170148786, −6.46629586391283543646346646067, −4.03800118799020079338147600589, −2.01549198704809618541833142767, 0.51457261522676747015917469618, 3.23214193338561395400116646262, 4.97523076659672317495670699754, 6.67674772481399100335420585473, 8.582337220508402254414083829671, 9.748027928008219967675131228300, 10.97934189533686084441418889251, 12.80498828864842461720506794851, 13.33737476709256844314061337516, 15.38442755022061614886999782615

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