Properties

Label 2-2e5-32.13-c7-0-16
Degree $2$
Conductor $32$
Sign $0.611 - 0.791i$
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  + (7.22 + 8.70i)2-s + (6.18 − 14.9i)3-s + (−23.4 + 125. i)4-s + (324. − 134. i)5-s + (174. − 54.1i)6-s + (441. − 441. i)7-s + (−1.26e3 + 705. i)8-s + (1.36e3 + 1.36e3i)9-s + (3.51e3 + 1.85e3i)10-s + (668. + 1.61e3i)11-s + (1.73e3 + 1.12e3i)12-s + (2.81e3 + 1.16e3i)13-s + (7.03e3 + 650. i)14-s − 5.67e3i·15-s + (−1.52e4 − 5.90e3i)16-s + 1.44e4i·17-s + ⋯
L(s)  = 1  + (0.638 + 0.769i)2-s + (0.132 − 0.319i)3-s + (−0.183 + 0.983i)4-s + (1.16 − 0.480i)5-s + (0.330 − 0.102i)6-s + (0.486 − 0.486i)7-s + (−0.873 + 0.487i)8-s + (0.622 + 0.622i)9-s + (1.11 + 0.585i)10-s + (0.151 + 0.365i)11-s + (0.289 + 0.188i)12-s + (0.354 + 0.146i)13-s + (0.685 + 0.0633i)14-s − 0.434i·15-s + (−0.932 − 0.360i)16-s + 0.711i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.611 - 0.791i$
Analytic conductor: \(9.99632\)
Root analytic conductor: \(3.16169\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :7/2),\ 0.611 - 0.791i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.74147 + 1.34707i\)
\(L(\frac12)\) \(\approx\) \(2.74147 + 1.34707i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.22 - 8.70i)T \)
good3 \( 1 + (-6.18 + 14.9i)T + (-1.54e3 - 1.54e3i)T^{2} \)
5 \( 1 + (-324. + 134. i)T + (5.52e4 - 5.52e4i)T^{2} \)
7 \( 1 + (-441. + 441. i)T - 8.23e5iT^{2} \)
11 \( 1 + (-668. - 1.61e3i)T + (-1.37e7 + 1.37e7i)T^{2} \)
13 \( 1 + (-2.81e3 - 1.16e3i)T + (4.43e7 + 4.43e7i)T^{2} \)
17 \( 1 - 1.44e4iT - 4.10e8T^{2} \)
19 \( 1 + (-7.86e3 - 3.25e3i)T + (6.32e8 + 6.32e8i)T^{2} \)
23 \( 1 + (2.27e4 + 2.27e4i)T + 3.40e9iT^{2} \)
29 \( 1 + (-8.77e4 + 2.11e5i)T + (-1.21e10 - 1.21e10i)T^{2} \)
31 \( 1 + 1.47e5T + 2.75e10T^{2} \)
37 \( 1 + (2.26e5 - 9.37e4i)T + (6.71e10 - 6.71e10i)T^{2} \)
41 \( 1 + (2.08e5 + 2.08e5i)T + 1.94e11iT^{2} \)
43 \( 1 + (3.15e5 + 7.60e5i)T + (-1.92e11 + 1.92e11i)T^{2} \)
47 \( 1 + 1.10e6iT - 5.06e11T^{2} \)
53 \( 1 + (-7.88e5 - 1.90e6i)T + (-8.30e11 + 8.30e11i)T^{2} \)
59 \( 1 + (2.03e6 - 8.43e5i)T + (1.75e12 - 1.75e12i)T^{2} \)
61 \( 1 + (5.94e5 - 1.43e6i)T + (-2.22e12 - 2.22e12i)T^{2} \)
67 \( 1 + (2.14e5 - 5.16e5i)T + (-4.28e12 - 4.28e12i)T^{2} \)
71 \( 1 + (1.56e6 - 1.56e6i)T - 9.09e12iT^{2} \)
73 \( 1 + (-4.37e5 - 4.37e5i)T + 1.10e13iT^{2} \)
79 \( 1 + 4.34e6iT - 1.92e13T^{2} \)
83 \( 1 + (5.99e6 + 2.48e6i)T + (1.91e13 + 1.91e13i)T^{2} \)
89 \( 1 + (7.82e5 - 7.82e5i)T - 4.42e13iT^{2} \)
97 \( 1 + 4.43e6T + 8.07e13T^{2} \)
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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

−15.34765459179879181304837987457, −13.88573334101211889954894095019, −13.45125410152939905863032817284, −12.20725569652575074800356444747, −10.25490654343597095066065199574, −8.614919238962348260835871404465, −7.25746289890984771395093899301, −5.78818568939902764955084929668, −4.36735918536527778463325535171, −1.84700120845042883898472675460, 1.55960611780638816728189902666, 3.21478240921639858736192134824, 5.12432406309236670621304850389, 6.47304870577626532825234066433, 9.116609893025749748067945296858, 10.08489316116065338898584656578, 11.30283508835827111846712154550, 12.67922570619611829087321553913, 13.91880229222048530801712506421, 14.69418558200616857605655937231

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