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Properties

Label 2-26e2-13.10-c1-0-0
Degree $2$
Conductor $676$
Sign $-0.998 - 0.0502i$
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  + (−0.678 − 1.17i)3-s + 4.24i·5-s + (−1.85 − 1.06i)7-s + (0.579 − 1.00i)9-s + (−2.90 + 1.67i)11-s + (4.99 − 2.88i)15-s + (−0.0304 + 0.0528i)17-s + (−5.35 − 3.09i)19-s + 2.89i·21-s + (−2.54 − 4.41i)23-s − 13.0·25-s − 5.64·27-s + (1.06 + 1.85i)29-s − 2.85i·31-s + (3.94 + 2.27i)33-s + ⋯
L(s)  = 1  + (−0.391 − 0.678i)3-s + 1.89i·5-s + (−0.699 − 0.403i)7-s + (0.193 − 0.334i)9-s + (−0.876 + 0.506i)11-s + (1.28 − 0.743i)15-s + (−0.00739 + 0.0128i)17-s + (−1.22 − 0.709i)19-s + 0.632i·21-s + (−0.531 − 0.920i)23-s − 2.60·25-s − 1.08·27-s + (0.198 + 0.343i)29-s − 0.512i·31-s + (0.686 + 0.396i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.998 - 0.0502i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ -0.998 - 0.0502i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00215696 + 0.0857435i\)
\(L(\frac12)\) \(\approx\) \(0.00215696 + 0.0857435i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.678 + 1.17i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 4.24iT - 5T^{2} \)
7 \( 1 + (1.85 + 1.06i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.90 - 1.67i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.0304 - 0.0528i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.35 + 3.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.54 + 4.41i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.06 - 1.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.85iT - 31T^{2} \)
37 \( 1 + (3.00 - 1.73i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.67 + 2.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.30 - 7.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.96iT - 47T^{2} \)
53 \( 1 - 0.0217T + 53T^{2} \)
59 \( 1 + (5.03 + 2.90i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.33 - 9.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.41 - 3.70i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.0423 + 0.0244i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.51iT - 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 - 3.55iT - 83T^{2} \)
89 \( 1 + (2.03 - 1.17i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.96 - 1.71i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.72960207507964361163588709300, −10.34900908377424528183727562494, −9.455956597075338218321725931521, −7.949096256895750734505755508818, −7.16216616024007258245422929005, −6.57278938597060690857435911308, −6.05108834112414667539396292831, −4.33663676564373778188914809364, −3.14921759042882999772691149264, −2.22368197636747809527039008892, 0.04433092865179348820119932724, 1.89654469956240736843450145690, 3.69621235740064540272327400913, 4.66810224316039627453442091127, 5.40280976461119957006518521615, 6.07432832080855333529534399324, 7.75303798281767756105048272409, 8.457564404239689888593673419466, 9.250779353180820071766701234469, 10.00422646831814578168824665362

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