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Properties

Label 2-2e10-16.13-c1-0-23
Degree $2$
Conductor $1024$
Sign $-0.382 + 0.923i$
Analytic cond. $8.17668$
Root an. cond. $2.85948$
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  + (1.41 − 1.41i)5-s − 3i·9-s + (−4.24 − 4.24i)13-s − 2·17-s + 0.999i·25-s + (−7.07 − 7.07i)29-s + (1.41 − 1.41i)37-s + 10i·41-s + (−4.24 − 4.24i)45-s + 7·49-s + (9.89 − 9.89i)53-s + (−7.07 − 7.07i)61-s − 12·65-s − 6i·73-s − 9·81-s + ⋯
L(s)  = 1  + (0.632 − 0.632i)5-s − i·9-s + (−1.17 − 1.17i)13-s − 0.485·17-s + 0.199i·25-s + (−1.31 − 1.31i)29-s + (0.232 − 0.232i)37-s + 1.56i·41-s + (−0.632 − 0.632i)45-s + 49-s + (1.35 − 1.35i)53-s + (−0.905 − 0.905i)61-s − 1.48·65-s − 0.702i·73-s − 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.311028777\)
\(L(\frac12)\) \(\approx\) \(1.311028777\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 3iT^{2} \)
5 \( 1 + (-1.41 + 1.41i)T - 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 11iT^{2} \)
13 \( 1 + (4.24 + 4.24i)T + 13iT^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (7.07 + 7.07i)T + 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-1.41 + 1.41i)T - 37iT^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-9.89 + 9.89i)T - 53iT^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + (7.07 + 7.07i)T + 61iT^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 10iT - 89T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−9.620283036764787944307423816710, −9.064602094731346530248000317992, −8.037174266388778325042618447529, −7.20843442642607362112930182312, −6.11847789407009324111669185490, −5.44051043365300152293167439512, −4.48202280828293009755043746083, −3.29094604743136645679953673768, −2.09309480223514768698021602910, −0.56167569235541526640622874703, 1.93156024107026740741843752164, 2.60007298966074515473555153780, 4.08713117179827191216354712982, 5.04205152256619503955879385253, 5.91652349504064375793631567283, 7.05859567686355949057528065175, 7.38519512176941067332095039794, 8.718483851383060738656172169502, 9.377663218754535898129479680363, 10.33337060286031108206974089336

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