Properties

Label 2-7e2-49.15-c3-0-8
Degree $2$
Conductor $49$
Sign $0.963 - 0.269i$
Analytic cond. $2.89109$
Root an. cond. $1.70032$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.892 + 3.90i)2-s + (5.02 − 6.30i)3-s + (−7.28 − 3.50i)4-s + (8.76 − 10.9i)5-s + (20.1 + 25.2i)6-s + (3.97 − 18.0i)7-s + (0.203 − 0.255i)8-s + (−8.46 − 37.1i)9-s + (35.1 + 44.0i)10-s + (−11.9 + 52.3i)11-s + (−58.7 + 28.2i)12-s + (−14.3 + 62.9i)13-s + (67.1 + 31.6i)14-s + (−25.2 − 110. i)15-s + (−39.4 − 49.5i)16-s + (47.5 − 22.9i)17-s + ⋯
L(s)  = 1  + (−0.315 + 1.38i)2-s + (0.967 − 1.21i)3-s + (−0.910 − 0.438i)4-s + (0.783 − 0.982i)5-s + (1.37 + 1.72i)6-s + (0.214 − 0.976i)7-s + (0.00899 − 0.0112i)8-s + (−0.313 − 1.37i)9-s + (1.11 + 1.39i)10-s + (−0.327 + 1.43i)11-s + (−1.41 + 0.680i)12-s + (−0.306 + 1.34i)13-s + (1.28 + 0.604i)14-s + (−0.434 − 1.90i)15-s + (−0.617 − 0.773i)16-s + (0.678 − 0.326i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.963 - 0.269i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ 0.963 - 0.269i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.61705 + 0.221766i\)
\(L(\frac12)\) \(\approx\) \(1.61705 + 0.221766i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-3.97 + 18.0i)T \)
good2 \( 1 + (0.892 - 3.90i)T + (-7.20 - 3.47i)T^{2} \)
3 \( 1 + (-5.02 + 6.30i)T + (-6.00 - 26.3i)T^{2} \)
5 \( 1 + (-8.76 + 10.9i)T + (-27.8 - 121. i)T^{2} \)
11 \( 1 + (11.9 - 52.3i)T + (-1.19e3 - 577. i)T^{2} \)
13 \( 1 + (14.3 - 62.9i)T + (-1.97e3 - 953. i)T^{2} \)
17 \( 1 + (-47.5 + 22.9i)T + (3.06e3 - 3.84e3i)T^{2} \)
19 \( 1 + 11.4T + 6.85e3T^{2} \)
23 \( 1 + (-89.1 - 42.9i)T + (7.58e3 + 9.51e3i)T^{2} \)
29 \( 1 + (38.3 - 18.4i)T + (1.52e4 - 1.90e4i)T^{2} \)
31 \( 1 + 67.2T + 2.97e4T^{2} \)
37 \( 1 + (386. - 185. i)T + (3.15e4 - 3.96e4i)T^{2} \)
41 \( 1 + (-28.5 + 35.7i)T + (-1.53e4 - 6.71e4i)T^{2} \)
43 \( 1 + (-122. - 153. i)T + (-1.76e4 + 7.75e4i)T^{2} \)
47 \( 1 + (-126. + 554. i)T + (-9.35e4 - 4.50e4i)T^{2} \)
53 \( 1 + (136. + 65.7i)T + (9.28e4 + 1.16e5i)T^{2} \)
59 \( 1 + (-107. - 134. i)T + (-4.57e4 + 2.00e5i)T^{2} \)
61 \( 1 + (-4.52 + 2.18i)T + (1.41e5 - 1.77e5i)T^{2} \)
67 \( 1 + 196.T + 3.00e5T^{2} \)
71 \( 1 + (71.8 + 34.5i)T + (2.23e5 + 2.79e5i)T^{2} \)
73 \( 1 + (160. + 701. i)T + (-3.50e5 + 1.68e5i)T^{2} \)
79 \( 1 - 241.T + 4.93e5T^{2} \)
83 \( 1 + (-93.8 - 411. i)T + (-5.15e5 + 2.48e5i)T^{2} \)
89 \( 1 + (139. + 612. i)T + (-6.35e5 + 3.05e5i)T^{2} \)
97 \( 1 + 1.68e3T + 9.12e5T^{2} \)
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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

−14.96976445762371366930490727693, −14.05288692463826652146665767455, −13.35962159240650968632946627302, −12.19964231888301491415097942470, −9.650114690577092719975859264669, −8.709034862407048202865779073579, −7.43829486051179202842311497010, −6.89858669083605483211371461588, −4.98439086880969148929531898753, −1.72638336240679091633733655690, 2.63180069809304397752291634513, 3.29977626535258508943422292299, 5.66246762652547143841165001490, 8.413320698751393202419154765911, 9.384599779911717281951441137914, 10.45299765962247563855822613408, 10.92235248732793822535911641319, 12.63126802131825980079392289761, 14.00807908763452670561776288242, 14.93740206576813839287556494113

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