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Properties

Label 2-7e2-7.2-c3-0-2
Degree $2$
Conductor $49$
Sign $-0.386 - 0.922i$
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  + (−1 + 1.73i)2-s + (3.5 + 6.06i)3-s + (2.00 + 3.46i)4-s + (3.5 − 6.06i)5-s − 14·6-s − 24·8-s + (−11 + 19.0i)9-s + (7 + 12.1i)10-s + (2.5 + 4.33i)11-s + (−13.9 + 24.2i)12-s + 14·13-s + 49·15-s + (8.00 − 13.8i)16-s + (−10.5 − 18.1i)17-s + (−21.9 − 38.1i)18-s + (24.5 − 42.4i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.673 + 1.16i)3-s + (0.250 + 0.433i)4-s + (0.313 − 0.542i)5-s − 0.952·6-s − 1.06·8-s + (−0.407 + 0.705i)9-s + (0.221 + 0.383i)10-s + (0.0685 + 0.118i)11-s + (−0.336 + 0.583i)12-s + 0.298·13-s + 0.843·15-s + (0.125 − 0.216i)16-s + (−0.149 − 0.259i)17-s + (−0.288 − 0.498i)18-s + (0.295 − 0.512i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.822291 + 1.23619i\)
\(L(\frac12)\) \(\approx\) \(0.822291 + 1.23619i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1 - 1.73i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-3.5 - 6.06i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-3.5 + 6.06i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 14T + 2.19e3T^{2} \)
17 \( 1 + (10.5 + 18.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-24.5 + 42.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-79.5 + 137. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 58T + 2.43e4T^{2} \)
31 \( 1 + (-73.5 - 127. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (109.5 - 189. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 350T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (-262.5 + 454. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (151.5 + 262. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (52.5 + 90.9i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (206.5 - 357. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (207.5 + 359. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 432T + 3.57e5T^{2} \)
73 \( 1 + (556.5 + 963. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-51.5 + 89.2i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.09e3T + 5.71e5T^{2} \)
89 \( 1 + (164.5 - 284. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 882T + 9.12e5T^{2} \)
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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

−15.57618236274701173286316897747, −14.77966723689247174771296786909, −13.37551521284181927205018725931, −11.98057225477039222686900033933, −10.39845376101178218881039258488, −9.088848293227599325778248676361, −8.507602741994005857856066353656, −6.79962182325509169258085329470, −4.87482792599805007294665219114, −3.15495962223867153442379067193, 1.48202847095998175464269468530, 2.89647039959383912277523104768, 6.02964880306457286738376039210, 7.22913365859080400240285678282, 8.686210137971942493694318853106, 10.02328686617267129357396298576, 11.21910479427611558524128774496, 12.40958691899981297245741830856, 13.63368336898788013741006311956, 14.47954291492294004914385608361

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