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Properties

Label 2-1274-91.4-c1-0-7
Degree $2$
Conductor $1274$
Sign $-0.302 + 0.953i$
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 + (1.67 + 2.89i)3-s − 4-s + (−1.35 + 0.781i)5-s + (−2.89 + 1.67i)6-s − i·8-s + (−4.09 + 7.09i)9-s + (−0.781 − 1.35i)10-s + (2.48 − 1.43i)11-s + (−1.67 − 2.89i)12-s + (−2.99 − 2.00i)13-s + (−4.52 − 2.61i)15-s + 16-s − 2.22·17-s + (−7.09 − 4.09i)18-s + (−6.26 − 3.61i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.965 + 1.67i)3-s − 0.5·4-s + (−0.605 + 0.349i)5-s + (−1.18 + 0.682i)6-s − 0.353i·8-s + (−1.36 + 2.36i)9-s + (−0.247 − 0.428i)10-s + (0.748 − 0.432i)11-s + (−0.482 − 0.836i)12-s + (−0.830 − 0.556i)13-s + (−1.16 − 0.675i)15-s + 0.250·16-s − 0.540·17-s + (−1.67 − 0.964i)18-s + (−1.43 − 0.830i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.302 + 0.953i$
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: \((2,\ 1274,\ (\ :1/2),\ -0.302 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.064713234\)
\(L(\frac12)\) \(\approx\) \(1.064713234\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.99 + 2.00i)T \)
good3 \( 1 + (-1.67 - 2.89i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.35 - 0.781i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.48 + 1.43i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.22T + 17T^{2} \)
19 \( 1 + (6.26 + 3.61i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.66T + 23T^{2} \)
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 \( 1 + 0.0385iT - 37T^{2} \)
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 \( 1 + 0.896iT - 59T^{2} \)
61 \( 1 + (7.12 - 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.42 + 0.820i)T + (33.5 - 58.0i)T^{2} \)
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 \( 1 - 2.42iT - 89T^{2} \)
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

−9.984779299861506073645414569396, −9.259522981149850411520511194411, −8.739012785018301595841163652349, −7.980773600691044960350768067720, −7.17116685029579639918975781483, −6.00125018401209691841317360376, −4.87950047934572727629398546922, −4.30987491362106277595855882227, −3.47415396873431550618135381980, −2.57996605880646755467975503037, 0.37378734200089260904200011276, 1.82155913157931239014144204305, 2.33860726430756934819444632911, 3.68414141628060768001608074669, 4.39394885879173190535914929878, 6.02184628196403003865943867224, 6.84542971250160307547350095098, 7.60130065721721836604409841979, 8.320522012552917418042977160822, 8.983090015433822190684437581966

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