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Properties

Label 2-1274-91.25-c1-0-42
Degree $2$
Conductor $1274$
Sign $-0.925 + 0.378i$
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  + (0.866 − 0.5i)2-s + (1.20 − 2.09i)3-s + (0.499 − 0.866i)4-s + (−1.22 + 0.707i)5-s − 2.41i·6-s − 0.999i·8-s + (−1.41 − 2.44i)9-s + (−0.707 + 1.22i)10-s + (−4.54 − 2.62i)11-s + (−1.20 − 2.09i)12-s + (3 − 2i)13-s + 3.41i·15-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (−2.44 − 1.41i)18-s + (0.210 − 0.121i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.696 − 1.20i)3-s + (0.249 − 0.433i)4-s + (−0.547 + 0.316i)5-s − 0.985i·6-s − 0.353i·8-s + (−0.471 − 0.816i)9-s + (−0.223 + 0.387i)10-s + (−1.36 − 0.790i)11-s + (−0.348 − 0.603i)12-s + (0.832 − 0.554i)13-s + 0.881i·15-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (−0.577 − 0.333i)18-s + (0.0482 − 0.0278i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.925 + 0.378i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.925 + 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.274849944\)
\(L(\frac12)\) \(\approx\) \(2.274849944\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 \)
13 \( 1 + (-3 + 2i)T \)
good3 \( 1 + (-1.20 + 2.09i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.22 - 0.707i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.54 + 2.62i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.210 + 0.121i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.792 - 1.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.41T + 29T^{2} \)
31 \( 1 + (4.54 + 2.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.30 - 1.32i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9iT - 41T^{2} \)
43 \( 1 - 3.75T + 43T^{2} \)
47 \( 1 + (3.82 - 2.20i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.707 - 1.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.87 - 5.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.1 - 6.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.75iT - 71T^{2} \)
73 \( 1 + (-11.6 - 6.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.86 + 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.58iT - 83T^{2} \)
89 \( 1 + (-9.79 + 5.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.48iT - 97T^{2} \)
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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

−9.192019569100193966361729088520, −8.265106367420392548000928124347, −7.52656524312386479815184144987, −7.16896420134288135277180017143, −5.82526564032766607095419831917, −5.27687741699718145644127433808, −3.64266175876248658669802552466, −3.07345714665724409133409360635, −2.11611840769720655637247115400, −0.69354644684877042286805953488, 2.10550422174978301402244359158, 3.44453449994589605098146518534, 3.93369696596389691733391047380, 4.80720374071893923037112823926, 5.54143933791673153959546361341, 6.70814044038743740111502697312, 7.905255581829928998768044341493, 8.223371000150486651543646098280, 9.190373098215917240508891554878, 10.01104520866267061486892000073

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