Properties

Label 2-7e2-49.8-c3-0-9
Degree $2$
Conductor $49$
Sign $-0.645 + 0.763i$
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  + (−2.45 − 3.07i)2-s + (2.79 − 1.34i)3-s + (−1.66 + 7.30i)4-s + (16.4 − 7.93i)5-s + (−10.9 − 5.28i)6-s + (−6.70 − 17.2i)7-s + (−1.80 + 0.871i)8-s + (−10.8 + 13.6i)9-s + (−64.8 − 31.2i)10-s + (6.93 + 8.69i)11-s + (5.16 + 22.6i)12-s + (−20.0 − 25.1i)13-s + (−36.6 + 62.9i)14-s + (35.3 − 44.3i)15-s + (61.1 + 29.4i)16-s + (−15.1 − 66.5i)17-s + ⋯
L(s)  = 1  + (−0.867 − 1.08i)2-s + (0.536 − 0.258i)3-s + (−0.208 + 0.912i)4-s + (1.47 − 0.710i)5-s + (−0.747 − 0.359i)6-s + (−0.361 − 0.932i)7-s + (−0.0799 + 0.0385i)8-s + (−0.402 + 0.504i)9-s + (−2.05 − 0.988i)10-s + (0.190 + 0.238i)11-s + (0.124 + 0.544i)12-s + (−0.427 − 0.535i)13-s + (−0.700 + 1.20i)14-s + (0.608 − 0.762i)15-s + (0.954 + 0.459i)16-s + (−0.216 − 0.949i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.645 + 0.763i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ -0.645 + 0.763i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.484878 - 1.04484i\)
\(L(\frac12)\) \(\approx\) \(0.484878 - 1.04484i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (6.70 + 17.2i)T \)
good2 \( 1 + (2.45 + 3.07i)T + (-1.78 + 7.79i)T^{2} \)
3 \( 1 + (-2.79 + 1.34i)T + (16.8 - 21.1i)T^{2} \)
5 \( 1 + (-16.4 + 7.93i)T + (77.9 - 97.7i)T^{2} \)
11 \( 1 + (-6.93 - 8.69i)T + (-296. + 1.29e3i)T^{2} \)
13 \( 1 + (20.0 + 25.1i)T + (-488. + 2.14e3i)T^{2} \)
17 \( 1 + (15.1 + 66.5i)T + (-4.42e3 + 2.13e3i)T^{2} \)
19 \( 1 - 113.T + 6.85e3T^{2} \)
23 \( 1 + (33.6 - 147. i)T + (-1.09e4 - 5.27e3i)T^{2} \)
29 \( 1 + (-40.8 - 179. i)T + (-2.19e4 + 1.05e4i)T^{2} \)
31 \( 1 - 167.T + 2.97e4T^{2} \)
37 \( 1 + (56.8 + 249. i)T + (-4.56e4 + 2.19e4i)T^{2} \)
41 \( 1 + (-214. + 103. i)T + (4.29e4 - 5.38e4i)T^{2} \)
43 \( 1 + (-81.3 - 39.1i)T + (4.95e4 + 6.21e4i)T^{2} \)
47 \( 1 + (-220. - 276. i)T + (-2.31e4 + 1.01e5i)T^{2} \)
53 \( 1 + (51.4 - 225. i)T + (-1.34e5 - 6.45e4i)T^{2} \)
59 \( 1 + (215. + 103. i)T + (1.28e5 + 1.60e5i)T^{2} \)
61 \( 1 + (7.36 + 32.2i)T + (-2.04e5 + 9.84e4i)T^{2} \)
67 \( 1 + 505.T + 3.00e5T^{2} \)
71 \( 1 + (152. - 666. i)T + (-3.22e5 - 1.55e5i)T^{2} \)
73 \( 1 + (-366. + 459. i)T + (-8.65e4 - 3.79e5i)T^{2} \)
79 \( 1 + 50.1T + 4.93e5T^{2} \)
83 \( 1 + (-348. + 437. i)T + (-1.27e5 - 5.57e5i)T^{2} \)
89 \( 1 + (730. - 915. i)T + (-1.56e5 - 6.87e5i)T^{2} \)
97 \( 1 - 5.56T + 9.12e5T^{2} \)
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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

−14.12345158815450814022636864223, −13.53068839178751961296981677137, −12.35324067019877407184754809933, −10.83209328677430640486443517236, −9.713692145045595376976005224746, −9.169830634872143963602783904679, −7.57726506767287956927086250007, −5.45143531769039461404640501922, −2.79293207738768927686340020412, −1.25142546321271291338610560729, 2.74236424569100957374869731312, 5.90364134770564412261242256087, 6.55715097602727783806791837531, 8.398005623348579564060269010443, 9.375760499435190062002058863448, 10.02695885005472655444758062537, 12.06215102358505770058092861434, 13.79625300173272047873533694568, 14.66495973865871552003406249037, 15.46110021059769741177545000720

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