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Properties

Label 2-7e2-7.4-c3-0-7
Degree $2$
Conductor $49$
Sign $-0.900 - 0.435i$
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  + (−2.26 − 3.92i)2-s + (1.78 − 3.09i)3-s + (−6.26 + 10.8i)4-s + (−6.73 − 11.6i)5-s − 16.2·6-s + 20.5·8-s + (7.09 + 12.2i)9-s + (−30.5 + 52.8i)10-s + (−0.406 + 0.704i)11-s + (22.4 + 38.8i)12-s − 34.9·13-s − 48.2·15-s + (3.60 + 6.25i)16-s + (58.8 − 101. i)17-s + (32.1 − 55.6i)18-s + (−46.6 − 80.7i)19-s + ⋯
L(s)  = 1  + (−0.800 − 1.38i)2-s + (0.344 − 0.596i)3-s + (−0.783 + 1.35i)4-s + (−0.602 − 1.04i)5-s − 1.10·6-s + 0.907·8-s + (0.262 + 0.455i)9-s + (−0.965 + 1.67i)10-s + (−0.0111 + 0.0193i)11-s + (0.539 + 0.934i)12-s − 0.745·13-s − 0.830·15-s + (0.0564 + 0.0976i)16-s + (0.839 − 1.45i)17-s + (0.420 − 0.728i)18-s + (−0.563 − 0.975i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.162400 + 0.707974i\)
\(L(\frac12)\) \(\approx\) \(0.162400 + 0.707974i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (2.26 + 3.92i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-1.78 + 3.09i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (6.73 + 11.6i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (0.406 - 0.704i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 34.9T + 2.19e3T^{2} \)
17 \( 1 + (-58.8 + 101. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (46.6 + 80.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (60.1 + 104. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 8.56T + 2.43e4T^{2} \)
31 \( 1 + (41.0 - 71.1i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (14.4 + 24.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 70.5T + 6.89e4T^{2} \)
43 \( 1 - 417.T + 7.95e4T^{2} \)
47 \( 1 + (-169. - 292. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (74.5 - 129. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (47.0 - 81.5i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (60.2 + 104. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-396. + 686. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 449.T + 3.57e5T^{2} \)
73 \( 1 + (234. - 406. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-509. - 883. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 104.T + 5.71e5T^{2} \)
89 \( 1 + (786. + 1.36e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 550.T + 9.12e5T^{2} \)
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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

−14.04879187052592171737994462988, −12.65845424850699331729412506518, −12.26303934805550370793851011107, −10.96263279384935511287514593301, −9.603831141810007269052954106901, −8.553249858518069419942135341582, −7.49151102814695637853186084996, −4.62711298756083186187153665339, −2.52407208301227235143754236447, −0.71187396613198385835126615551, 3.74036059967678764411312221647, 5.95731012561353946710290128700, 7.24184998822917238753037461662, 8.204806772655303440351642672579, 9.617903893826372010795153350280, 10.49784718025184492732753465998, 12.26234005837065570976509958625, 14.40961028505509290669100907935, 14.89731227481053552121992675262, 15.60687440001039640497724586714

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