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Properties

Label 2-26e2-13.3-c1-0-3
Degree $2$
Conductor $676$
Sign $0.0128 - 0.999i$
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 + 1.73i)3-s + 3·5-s + (−2 + 3.46i)7-s + (−0.499 + 0.866i)9-s + (3 + 5.19i)15-s + (−1.5 + 2.59i)17-s + (1 − 1.73i)19-s − 7.99·21-s + (3 + 5.19i)23-s + 4·25-s + 4.00·27-s + (−4.5 − 7.79i)29-s − 2·31-s + (−6 + 10.3i)35-s + (−3.5 − 6.06i)37-s + ⋯
L(s)  = 1  + (0.577 + 0.999i)3-s + 1.34·5-s + (−0.755 + 1.30i)7-s + (−0.166 + 0.288i)9-s + (0.774 + 1.34i)15-s + (−0.363 + 0.630i)17-s + (0.229 − 0.397i)19-s − 1.74·21-s + (0.625 + 1.08i)23-s + 0.800·25-s + 0.769·27-s + (−0.835 − 1.44i)29-s − 0.359·31-s + (−1.01 + 1.75i)35-s + (−0.575 − 0.996i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47685 + 1.45804i\)
\(L(\frac12)\) \(\approx\) \(1.47685 + 1.45804i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
7 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.35560157465230223194996454498, −9.630453088540719006615741088672, −9.209257756324542347096255573778, −8.631328607182920934063792943016, −7.09528233438551949372320990265, −5.86975554491085492382673460408, −5.53909783371407478066925571533, −4.11263979243731739002034386271, −2.97597831775266201140342160886, −2.07711158243366803850872861099, 1.10678632525517700864777588364, 2.26959389558707272582976937652, 3.36708266486775525465195576908, 4.82725718464144349283243456518, 6.05537491102806553302358037419, 6.96112979680475448652084198151, 7.32374642384608213755406093035, 8.612550988928739262052648196416, 9.406321567730820213202728551975, 10.27101224286537867897798549883

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