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Properties

Label 2-26e2-13.10-c1-0-1
Degree $2$
Conductor $676$
Sign $-0.917 - 0.397i$
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  + (1.52 + 2.64i)3-s + 2.55i·5-s + (−2.73 − 1.57i)7-s + (−3.14 + 5.45i)9-s + (0.908 − 0.524i)11-s + (−6.74 + 3.89i)15-s + (−1.04 + 1.80i)17-s + (−2.13 − 1.23i)19-s − 9.63i·21-s + (2.19 + 3.79i)23-s − 1.52·25-s − 10.0·27-s + (1.57 + 2.73i)29-s + 4.93i·31-s + (2.76 + 1.59i)33-s + ⋯
L(s)  = 1  + (0.880 + 1.52i)3-s + 1.14i·5-s + (−1.03 − 0.596i)7-s + (−1.04 + 1.81i)9-s + (0.273 − 0.158i)11-s + (−1.74 + 1.00i)15-s + (−0.253 + 0.438i)17-s + (−0.490 − 0.282i)19-s − 2.10i·21-s + (0.457 + 0.791i)23-s − 0.305·25-s − 1.93·27-s + (0.293 + 0.507i)29-s + 0.887i·31-s + (0.482 + 0.278i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.917 - 0.397i$
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: \((2,\ 676,\ (\ :1/2),\ -0.917 - 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.316760 + 1.52935i\)
\(L(\frac12)\) \(\approx\) \(0.316760 + 1.52935i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 - 2.64i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.55iT - 5T^{2} \)
7 \( 1 + (2.73 + 1.57i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.908 + 0.524i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.04 - 1.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.13 + 1.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.19 - 3.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.57 - 2.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.93iT - 31T^{2} \)
37 \( 1 + (7.41 - 4.28i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.95 + 5.74i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.36 + 11.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.54iT - 47T^{2} \)
53 \( 1 + 8.45T + 53T^{2} \)
59 \( 1 + (-10.8 - 6.24i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.78 - 3.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.50 + 0.870i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.06 - 2.34i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.34iT - 73T^{2} \)
79 \( 1 - 6.73T + 79T^{2} \)
83 \( 1 - 2.19iT - 83T^{2} \)
89 \( 1 + (11.4 - 6.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.4 - 6.62i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.51136142945925756725650589748, −10.19924194238625560744023072580, −9.217139129375962212128737719417, −8.627701492167265322323185459654, −7.30408229168919780948725681982, −6.57742671194834319391003583053, −5.28219647382518339235981616740, −3.96237228978306739484432588518, −3.46782066508672240971864288146, −2.57396803605795280829615057478, 0.73773220244554931826043531821, 2.11073157619763942821038577747, 3.07305144309109856001989011874, 4.50391364530443526612662398365, 5.97279242545643759952343746680, 6.56476561204863665194522246090, 7.62039500734037465961738515296, 8.394624473059717575883411416150, 9.121508238745262797759970207531, 9.603796610819466882716100881997

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