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Properties

Label 2-1274-91.30-c1-0-38
Degree $2$
Conductor $1274$
Sign $0.762 + 0.647i$
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 + 3.34·3-s + (0.499 − 0.866i)4-s + (−1.35 − 0.781i)5-s + (2.89 − 1.67i)6-s − 0.999i·8-s + 8.18·9-s − 1.56·10-s + 2.86i·11-s + (1.67 − 2.89i)12-s + (2.99 + 2.00i)13-s + (−4.52 − 2.61i)15-s + (−0.5 − 0.866i)16-s + (−1.11 + 1.93i)17-s + (7.09 − 4.09i)18-s − 7.23i·19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + 1.93·3-s + (0.249 − 0.433i)4-s + (−0.605 − 0.349i)5-s + (1.18 − 0.682i)6-s − 0.353i·8-s + 2.72·9-s − 0.494·10-s + 0.864i·11-s + (0.482 − 0.836i)12-s + (0.830 + 0.556i)13-s + (−1.16 − 0.675i)15-s + (−0.125 − 0.216i)16-s + (−0.270 + 0.468i)17-s + (1.67 − 0.964i)18-s − 1.66i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $0.762 + 0.647i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ 0.762 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.297187245\)
\(L(\frac12)\) \(\approx\) \(4.297187245\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.99 - 2.00i)T \)
good3 \( 1 - 3.34T + 3T^{2} \)
5 \( 1 + (1.35 + 0.781i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 2.86iT - 11T^{2} \)
17 \( 1 + (1.11 - 1.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7.23iT - 19T^{2} \)
23 \( 1 + (0.833 + 1.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.41 - 4.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.517 - 0.298i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0333 - 0.0192i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.88 + 3.97i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.04 - 8.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.08 + 3.51i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.99 - 5.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.776 + 0.448i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 - 1.64iT - 67T^{2} \)
71 \( 1 + (1.98 - 1.14i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.72 + 5.61i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.13 - 3.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.94iT - 83T^{2} \)
89 \( 1 + (2.09 - 1.21i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.23 - 2.44i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.323284313400141476320959010217, −8.876771628794004557278771167413, −8.073206224625199843390911155590, −7.23832134723589687194308180618, −6.52476850090281475746949075567, −4.76898911227489926036561844182, −4.23640637827683892156606084572, −3.43444461792412586397002472822, −2.46269002819258438135014697431, −1.52690760359138421053551007120, 1.72018730311391758341025805891, 3.01179980379330173148091155962, 3.56624936573863701633210574540, 4.14615209044207040600742754243, 5.58563757443340050830284273753, 6.63440329555470977191464532598, 7.65166484804833525514212353680, 8.007599255679866817316193895339, 8.662756771132806528683051926074, 9.557246420153453747707413215606

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