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Properties

Label 2-1274-91.4-c1-0-36
Degree $2$
Conductor $1274$
Sign $-0.408 + 0.912i$
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 + (−0.233 − 0.404i)3-s − 4-s + (2.93 − 1.69i)5-s + (−0.404 + 0.233i)6-s + i·8-s + (1.39 − 2.40i)9-s + (−1.69 − 2.93i)10-s + (−0.712 + 0.411i)11-s + (0.233 + 0.404i)12-s + (2.74 + 2.33i)13-s + (−1.36 − 0.790i)15-s + 16-s + 4.58·17-s + (−2.40 − 1.39i)18-s + (−5.11 − 2.95i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.134 − 0.233i)3-s − 0.5·4-s + (1.31 − 0.757i)5-s + (−0.164 + 0.0952i)6-s + 0.353i·8-s + (0.463 − 0.803i)9-s + (−0.535 − 0.928i)10-s + (−0.214 + 0.124i)11-s + (0.0673 + 0.116i)12-s + (0.762 + 0.646i)13-s + (−0.353 − 0.204i)15-s + 0.250·16-s + 1.11·17-s + (−0.567 − 0.327i)18-s + (−1.17 − 0.677i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.408 + 0.912i$
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: \((2,\ 1274,\ (\ :1/2),\ -0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.023742032\)
\(L(\frac12)\) \(\approx\) \(2.023742032\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.74 - 2.33i)T \)
good3 \( 1 + (0.233 + 0.404i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.93 + 1.69i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.712 - 0.411i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 4.58T + 17T^{2} \)
19 \( 1 + (5.11 + 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.13T + 23T^{2} \)
29 \( 1 + (-3.43 + 5.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.71 + 2.14i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.69iT - 37T^{2} \)
41 \( 1 + (-0.0774 - 0.0446i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.67 - 6.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.67 - 5.58i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.50 + 6.07i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.74iT - 59T^{2} \)
61 \( 1 + (-1.18 + 2.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.252 - 0.145i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.48 + 5.47i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (11.0 + 6.35i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.97 + 8.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.23iT - 83T^{2} \)
89 \( 1 - 8.04iT - 89T^{2} \)
97 \( 1 + (12.7 - 7.38i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.512478290464466542659112464203, −8.919435127636214684401078809363, −8.037348664336985023652192536294, −6.65774989404992082006896742645, −6.11041868816058797740479581926, −5.07877657985572752869698604663, −4.28229618392741288469632626757, −3.02399824541402958529613665541, −1.79465512198408676362678734196, −0.984925502894396503257727655376, 1.52901896220035686047992146154, 2.81510548413131007280558312458, 3.94779299609895163002801590612, 5.41032505892788331667475533668, 5.53990613953791688218508859950, 6.63649275712792019009476905111, 7.30756888757399960361707721003, 8.299036021803368492686735940914, 9.075998070198010390210512505041, 10.10772481505972220749665908011

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