Properties

Label 2-7e2-49.37-c1-0-1
Degree $2$
Conductor $49$
Sign $0.819 - 0.573i$
Analytic cond. $0.391266$
Root an. cond. $0.625513$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.968 + 0.659i)2-s + (0.630 − 0.584i)3-s + (−0.229 + 0.583i)4-s + (3.03 + 0.936i)5-s + (−0.224 + 0.982i)6-s + (−2.50 − 0.844i)7-s + (−0.684 − 3.00i)8-s + (−0.168 + 2.25i)9-s + (−3.55 + 1.09i)10-s + (−0.358 − 4.78i)11-s + (0.197 + 0.502i)12-s + (−3.44 − 1.65i)13-s + (2.98 − 0.836i)14-s + (2.46 − 1.18i)15-s + (1.72 + 1.59i)16-s + (0.602 + 0.0908i)17-s + ⋯
L(s)  = 1  + (−0.684 + 0.466i)2-s + (0.363 − 0.337i)3-s + (−0.114 + 0.291i)4-s + (1.35 + 0.418i)5-s + (−0.0915 + 0.400i)6-s + (−0.947 − 0.319i)7-s + (−0.242 − 1.06i)8-s + (−0.0563 + 0.751i)9-s + (−1.12 + 0.346i)10-s + (−0.108 − 1.44i)11-s + (0.0568 + 0.144i)12-s + (−0.955 − 0.460i)13-s + (0.797 − 0.223i)14-s + (0.635 − 0.305i)15-s + (0.430 + 0.399i)16-s + (0.146 + 0.0220i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.819 - 0.573i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :1/2),\ 0.819 - 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656378 + 0.206949i\)
\(L(\frac12)\) \(\approx\) \(0.656378 + 0.206949i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.50 + 0.844i)T \)
good2 \( 1 + (0.968 - 0.659i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-0.630 + 0.584i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (-3.03 - 0.936i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (0.358 + 4.78i)T + (-10.8 + 1.63i)T^{2} \)
13 \( 1 + (3.44 + 1.65i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-0.602 - 0.0908i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (1.98 - 3.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.22 + 0.486i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (0.769 - 0.965i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-0.607 - 1.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.77 + 7.07i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-1.52 - 6.66i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.100 + 0.439i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (5.92 - 4.04i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-0.873 + 2.22i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-3.38 + 1.04i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-4.53 - 11.5i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-1.61 - 2.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.01 + 6.29i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-2.14 - 1.45i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (2.76 - 4.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.63 + 1.27i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-0.623 + 8.32i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 6.49T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.25892309100275763905753452861, −14.45625867274441921910644581930, −13.44431536995183012668148973981, −12.78592081737007342241226953391, −10.58123773484093621659543453302, −9.654379970155721774314448618822, −8.431671513665082207879619365142, −7.13739021528771034543487620064, −5.87624500497358214110217519885, −2.98385116821430465205076926101, 2.31367542381274949372315777704, 5.05289993446727918767253369310, 6.62121983070430960727250225797, 9.005173385777941671303412729947, 9.586644823742727031322313030340, 10.14818212600568737206435617503, 12.08506890883546630106506981927, 13.19607487213481000906458085349, 14.51179834043876496499623580813, 15.36378042725775037772710110698

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