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Properties

Label 2-1274-13.10-c1-0-0
Degree $2$
Conductor $1274$
Sign $-0.697 + 0.716i$
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 + (0.703 + 1.21i)3-s + (0.499 − 0.866i)4-s + 0.638i·5-s + (−1.21 − 0.703i)6-s + 0.999i·8-s + (0.511 − 0.885i)9-s + (−0.319 − 0.552i)10-s + (−3.93 + 2.27i)11-s + 1.40·12-s + (−2.48 + 2.61i)13-s + (−0.777 + 0.448i)15-s + (−0.5 − 0.866i)16-s + (−0.354 + 0.613i)17-s + 1.02i·18-s + (−6.36 − 3.67i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.405 + 0.703i)3-s + (0.249 − 0.433i)4-s + 0.285i·5-s + (−0.497 − 0.287i)6-s + 0.353i·8-s + (0.170 − 0.295i)9-s + (−0.100 − 0.174i)10-s + (−1.18 + 0.685i)11-s + 0.405·12-s + (−0.688 + 0.725i)13-s + (−0.200 + 0.115i)15-s + (−0.125 − 0.216i)16-s + (−0.0859 + 0.148i)17-s + 0.241i·18-s + (−1.45 − 0.842i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.697 + 0.716i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.697 + 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1961722106\)
\(L(\frac12)\) \(\approx\) \(0.1961722106\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.48 - 2.61i)T \)
good3 \( 1 + (-0.703 - 1.21i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.638iT - 5T^{2} \)
11 \( 1 + (3.93 - 2.27i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.354 - 0.613i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.36 + 3.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.25 + 5.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.63 + 6.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.63iT - 31T^{2} \)
37 \( 1 + (1.82 - 1.05i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.62 - 2.66i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.21 - 9.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.94iT - 47T^{2} \)
53 \( 1 + 6.40T + 53T^{2} \)
59 \( 1 + (2.87 + 1.65i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.41 + 5.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.39 + 1.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.11 + 5.26i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.41iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 2.41iT - 83T^{2} \)
89 \( 1 + (-4.79 + 2.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.96 - 4.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.17306160462094761139411415639, −9.373228254277657772473628861904, −8.635355091624682765373171395106, −7.902761398487052067697116833316, −6.85427493670489361257656226379, −6.39205096112189645980219505187, −4.85413030262843781584008627988, −4.45061829864159195340488706682, −2.97136530395921577102264004903, −2.05191243552983658338279109522, 0.087402431783704131036516152146, 1.68737611478391034324437930332, 2.54263372825883147339425907469, 3.59345006900731779395518512686, 4.97650960499885325516670659633, 5.83362062542605735764590085146, 7.08626192827358403111713670298, 7.70631234662631967679502936139, 8.327848474069392907245631962158, 8.942225000570666085733933124299

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