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Properties

Label 2-7e2-7.4-c5-0-11
Degree $2$
Conductor $49$
Sign $0.900 + 0.435i$
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  + (3.90 + 6.76i)2-s + (11.7 − 20.3i)3-s + (−14.5 + 25.1i)4-s + (−37.1 − 64.3i)5-s + 183.·6-s + 22.8·8-s + (−155. − 268. i)9-s + (290. − 502. i)10-s + (212. − 367. i)11-s + (341. + 592. i)12-s + 252.·13-s − 1.74e3·15-s + (554. + 960. i)16-s + (−552. + 956. i)17-s + (1.21e3 − 2.09e3i)18-s + (−3.23 − 5.60i)19-s + ⋯
L(s)  = 1  + (0.690 + 1.19i)2-s + (0.754 − 1.30i)3-s + (−0.454 + 0.786i)4-s + (−0.664 − 1.15i)5-s + 2.08·6-s + 0.126·8-s + (−0.638 − 1.10i)9-s + (0.917 − 1.58i)10-s + (0.528 − 0.915i)11-s + (0.685 + 1.18i)12-s + 0.413·13-s − 2.00·15-s + (0.541 + 0.937i)16-s + (−0.463 + 0.802i)17-s + (0.881 − 1.52i)18-s + (−0.00205 − 0.00356i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.900 + 0.435i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ 0.900 + 0.435i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.68074 - 0.614927i\)
\(L(\frac12)\) \(\approx\) \(2.68074 - 0.614927i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-3.90 - 6.76i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (-11.7 + 20.3i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (37.1 + 64.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-212. + 367. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 252.T + 3.71e5T^{2} \)
17 \( 1 + (552. - 956. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (3.23 + 5.60i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.80e3 - 3.12e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 5.00e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.41e3 + 2.44e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-1.02e3 - 1.77e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 9.39e3T + 1.15e8T^{2} \)
43 \( 1 - 1.03e4T + 1.47e8T^{2} \)
47 \( 1 + (-8.51e3 - 1.47e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.97e4 + 3.42e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.69e4 + 2.94e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.41e4 - 2.45e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.80e4 - 4.85e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.55e4T + 1.80e9T^{2} \)
73 \( 1 + (3.91e4 - 6.77e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-2.26e4 - 3.92e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.38e3T + 3.93e9T^{2} \)
89 \( 1 + (3.44e4 + 5.96e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.08e5T + 8.58e9T^{2} \)
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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

−14.42977272541819611096262543363, −13.31687900690218385871098246436, −12.93801858137957319890732867469, −11.50678285493164396843333249235, −8.786291018067084283487083854452, −8.103682007317413868757946323518, −7.00373435371966113812312509052, −5.68517510031743784433705286655, −3.90959184547142977400297893082, −1.24743851160453769340770646466, 2.59258709503750789230589184514, 3.65959453145992713367181972330, 4.60010617073679749296822578628, 7.21185592742091114161484428915, 9.084314855225220842956144814581, 10.33207553657226152358126201921, 10.99701490523002165188333568287, 12.12224692251080996259268863708, 13.67679423875699072513971282699, 14.70302664399617234008411307514

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