Properties

Label 2-7e2-49.2-c5-0-6
Degree $2$
Conductor $49$
Sign $-0.713 - 0.700i$
Analytic cond. $7.85880$
Root an. cond. $2.80335$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.58 − 2.34i)2-s + (−8.77 + 22.3i)3-s + (25.6 + 17.4i)4-s + (44.1 − 6.65i)5-s + (118. − 149. i)6-s + (109. + 68.7i)7-s + (4.81 + 6.04i)8-s + (−245. − 227. i)9-s + (−350. − 52.8i)10-s + (−18.2 + 16.9i)11-s + (−616. + 419. i)12-s + (101. + 445. i)13-s + (−673. − 778. i)14-s + (−238. + 1.04e3i)15-s + (−385. − 981. i)16-s + (113. + 1.51e3i)17-s + ⋯
L(s)  = 1  + (−1.34 − 0.413i)2-s + (−0.563 + 1.43i)3-s + (0.801 + 0.546i)4-s + (0.790 − 0.119i)5-s + (1.34 − 1.69i)6-s + (0.847 + 0.530i)7-s + (0.0266 + 0.0333i)8-s + (−1.00 − 0.936i)9-s + (−1.10 − 0.167i)10-s + (−0.0454 + 0.0421i)11-s + (−1.23 + 0.841i)12-s + (0.166 + 0.731i)13-s + (−0.917 − 1.06i)14-s + (−0.274 + 1.20i)15-s + (−0.376 − 0.958i)16-s + (0.0955 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.713 - 0.700i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -0.713 - 0.700i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.241340 + 0.590692i\)
\(L(\frac12)\) \(\approx\) \(0.241340 + 0.590692i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-109. - 68.7i)T \)
good2 \( 1 + (7.58 + 2.34i)T + (26.4 + 18.0i)T^{2} \)
3 \( 1 + (8.77 - 22.3i)T + (-178. - 165. i)T^{2} \)
5 \( 1 + (-44.1 + 6.65i)T + (2.98e3 - 921. i)T^{2} \)
11 \( 1 + (18.2 - 16.9i)T + (1.20e4 - 1.60e5i)T^{2} \)
13 \( 1 + (-101. - 445. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (-113. - 1.51e3i)T + (-1.40e6 + 2.11e5i)T^{2} \)
19 \( 1 + (877. - 1.52e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-363. + 4.85e3i)T + (-6.36e6 - 9.59e5i)T^{2} \)
29 \( 1 + (945. + 455. i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (959. + 1.66e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (1.07e4 - 7.32e3i)T + (2.53e7 - 6.45e7i)T^{2} \)
41 \( 1 + (-4.28e3 - 5.37e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (-1.81e3 + 2.27e3i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (-1.14e4 - 3.53e3i)T + (1.89e8 + 1.29e8i)T^{2} \)
53 \( 1 + (-1.57e4 - 1.07e4i)T + (1.52e8 + 3.89e8i)T^{2} \)
59 \( 1 + (2.43e4 + 3.66e3i)T + (6.83e8 + 2.10e8i)T^{2} \)
61 \( 1 + (2.12e4 - 1.44e4i)T + (3.08e8 - 7.86e8i)T^{2} \)
67 \( 1 + (2.59e4 + 4.49e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-3.31e4 + 1.59e4i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (1.76e4 - 5.43e3i)T + (1.71e9 - 1.16e9i)T^{2} \)
79 \( 1 + (4.06e4 - 7.04e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-5.34e3 + 2.34e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (-8.52e4 - 7.90e4i)T + (4.17e8 + 5.56e9i)T^{2} \)
97 \( 1 + 1.22e5T + 8.58e9T^{2} \)
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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

−15.27563602760743745663239707571, −14.22531047200270242786276432520, −12.13321752720266139961771733833, −10.85273546484747479333011940744, −10.32265197178730583781471033651, −9.222900150671201339320075035595, −8.347227159106267655487224834834, −5.91428979192731971659905268838, −4.49020670537124217005146439060, −1.85500274482877869425792337862, 0.55946039944398722433331769586, 1.78038413227521479352387548628, 5.58110586478821299520967999192, 7.06289542335359430951714033460, 7.65239351007565454091756748021, 9.081752966209015220842311877367, 10.51923275347067743006791402666, 11.58360865558730948917661350959, 13.16611827286635515291961625301, 13.87597320131656649533845995113

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