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Properties

Label 2-1274-91.4-c1-0-15
Degree $2$
Conductor $1274$
Sign $-0.221 + 0.975i$
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  − i·2-s + (−1.27 − 2.21i)3-s − 4-s + (−3.02 + 1.74i)5-s + (−2.21 + 1.27i)6-s + i·8-s + (−1.76 + 3.06i)9-s + (1.74 + 3.02i)10-s + (2.32 − 1.34i)11-s + (1.27 + 2.21i)12-s + (3.15 + 1.74i)13-s + (7.72 + 4.45i)15-s + 16-s − 5.91·17-s + (3.06 + 1.76i)18-s + (4.50 + 2.59i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.737 − 1.27i)3-s − 0.5·4-s + (−1.35 + 0.779i)5-s + (−0.903 + 0.521i)6-s + 0.353i·8-s + (−0.589 + 1.02i)9-s + (0.551 + 0.955i)10-s + (0.700 − 0.404i)11-s + (0.368 + 0.639i)12-s + (0.874 + 0.484i)13-s + (1.99 + 1.15i)15-s + 0.250·16-s − 1.43·17-s + (0.721 + 0.416i)18-s + (1.03 + 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.221 + 0.975i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.221 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8323531931\)
\(L(\frac12)\) \(\approx\) \(0.8323531931\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
7 \( 1 \)
13 \( 1 + (-3.15 - 1.74i)T \)
good3 \( 1 + (1.27 + 2.21i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (3.02 - 1.74i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.32 + 1.34i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 5.91T + 17T^{2} \)
19 \( 1 + (-4.50 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.05T + 23T^{2} \)
29 \( 1 + (3.56 - 6.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.677 + 0.391i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.81iT - 37T^{2} \)
41 \( 1 + (0.136 + 0.0788i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.165 + 0.285i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.38 - 0.800i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.96 - 3.40i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.54iT - 59T^{2} \)
61 \( 1 + (-7.70 + 13.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.837 - 0.483i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.62 + 2.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-13.0 - 7.51i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.146 - 0.254i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.87iT - 83T^{2} \)
89 \( 1 - 5.21iT - 89T^{2} \)
97 \( 1 + (-2.73 + 1.57i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.358165747762349116710789504302, −8.584203943517940986875768688714, −7.65320490359435119005353572699, −6.93399031782204777394072356426, −6.42251755639968193168463246377, −5.23041442871504257311120714694, −3.94626274977477610897450526212, −3.27651277130700016643183707046, −1.83334200559304297586990346971, −0.66027470952254163942948482001, 0.76023693354136824915580623011, 3.40329931904238203796130778593, 4.22879304854545881928872236333, 4.71749788759975881675367873248, 5.48329804179565598288288309343, 6.57953166118411917089870075502, 7.43118004532410904664276556294, 8.448456291564425846384511678226, 9.029578523042112044515534665457, 9.699308066365292319453924826099

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