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Properties

Label 2-1274-7.2-c1-0-34
Degree $2$
Conductor $1274$
Sign $-0.991 - 0.126i$
Analytic cond. $10.1729$
Root an. cond. $3.18950$
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  + (−0.5 + 0.866i)2-s + (−1.5 − 2.59i)3-s + (−0.499 − 0.866i)4-s + (2 − 3.46i)5-s + 3·6-s + 0.999·8-s + (−3 + 5.19i)9-s + (1.99 + 3.46i)10-s + (−0.5 − 0.866i)11-s + (−1.50 + 2.59i)12-s − 13-s − 12·15-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)18-s + (3 − 5.19i)19-s − 3.99·20-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.866 − 1.49i)3-s + (−0.249 − 0.433i)4-s + (0.894 − 1.54i)5-s + 1.22·6-s + 0.353·8-s + (−1 + 1.73i)9-s + (0.632 + 1.09i)10-s + (−0.150 − 0.261i)11-s + (−0.433 + 0.749i)12-s − 0.277·13-s − 3.09·15-s + (−0.125 + 0.216i)16-s + (−0.707 − 1.22i)18-s + (0.688 − 1.19i)19-s − 0.894·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8070926367\)
\(L(\frac12)\) \(\approx\) \(0.8070926367\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
13 \( 1 + T \)
good3 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + (2.5 + 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−9.021594127433278379638206563697, −8.382568948786264269487885069334, −7.52797768626457675565164781243, −6.73565909885860207790899875734, −5.96928570800475926006589990578, −5.28574528877152935914182696175, −4.71016423673237050421320595122, −2.37640809178145175169787267256, −1.28424668194566112350330942710, −0.45543412837843330237426882154, 1.92614253494612465849687376707, 3.30293151351504200317268486616, 3.70039284966342558337052914706, 5.21295223087484650788380698945, 5.62392069734430319538721511653, 6.74344230799443447443386479398, 7.55132031901874429827242824087, 9.051353354120912711582417601800, 9.691263019391763073838050297185, 10.13371298581512180636247451432

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