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Properties

Label 2-26e2-13.4-c1-0-4
Degree $2$
Conductor $676$
Sign $0.431 - 0.902i$
Analytic cond. $5.39788$
Root an. cond. $2.32333$
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  + 2i·5-s + (1.73 − i)7-s + (1.5 + 2.59i)9-s + (−1.73 − i)11-s + (3 + 5.19i)17-s + (−5.19 + 3i)19-s + (4 − 6.92i)23-s + 25-s + (−1 + 1.73i)29-s + 10i·31-s + (2 + 3.46i)35-s + (−5.19 − 3i)37-s + (5.19 + 3i)41-s + (2 + 3.46i)43-s + (−5.19 + 3i)45-s + ⋯
L(s)  = 1  + 0.894i·5-s + (0.654 − 0.377i)7-s + (0.5 + 0.866i)9-s + (−0.522 − 0.301i)11-s + (0.727 + 1.26i)17-s + (−1.19 + 0.688i)19-s + (0.834 − 1.44i)23-s + 0.200·25-s + (−0.185 + 0.321i)29-s + 1.79i·31-s + (0.338 + 0.585i)35-s + (−0.854 − 0.493i)37-s + (0.811 + 0.468i)41-s + (0.304 + 0.528i)43-s + (−0.774 + 0.447i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.431 - 0.902i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ 0.431 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31518 + 0.828611i\)
\(L(\frac12)\) \(\approx\) \(1.31518 + 0.828611i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.73 + i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.19 - 3i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
37 \( 1 + (5.19 + 3i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.19 - 3i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-8.66 + 5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.66 + 5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.66 + 5i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (5.19 + 3i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.73 + i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.56376766077928565771738840086, −10.32482870950398781795881475873, −8.680479724165210270542305775826, −8.045510277133074787805051191262, −7.17286586144960115774683947045, −6.30578500370361918790686755788, −5.13913350775204915801230634840, −4.17089519120655235947662451445, −2.92652402557781693199800172786, −1.65945781419114502170589841976, 0.900296958495734420879309334325, 2.38747697654856283914929171576, 3.90046857655241661429296421437, 4.93202993518471286938340568819, 5.58834495006012869278686085225, 6.94396092471975609800147484398, 7.72143600870633797571159548253, 8.791650783051026625015528119548, 9.327991531441754986524133019239, 10.20746035875608559958966904941

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