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Properties

Label 2-7e2-7.4-c1-0-0
Degree $2$
Conductor $49$
Sign $0.605 + 0.795i$
Analytic cond. $0.391266$
Root an. cond. $0.625513$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s − 3·8-s + (1.5 + 2.59i)9-s + (−2 + 3.46i)11-s + (0.500 + 0.866i)16-s + (1.5 − 2.59i)18-s + 3.99·22-s + (−4 − 6.92i)23-s + (2.5 − 4.33i)25-s + 2·29-s + (−2.50 + 4.33i)32-s + 3·36-s + (3 + 5.19i)37-s − 12·43-s + (1.99 + 3.46i)44-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s − 1.06·8-s + (0.5 + 0.866i)9-s + (−0.603 + 1.04i)11-s + (0.125 + 0.216i)16-s + (0.353 − 0.612i)18-s + 0.852·22-s + (−0.834 − 1.44i)23-s + (0.5 − 0.866i)25-s + 0.371·29-s + (−0.441 + 0.765i)32-s + 0.5·36-s + (0.493 + 0.854i)37-s − 1.82·43-s + (0.301 + 0.522i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.654693 - 0.324549i\)
\(L(\frac12)\) \(\approx\) \(0.654693 - 0.324549i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−15.49241818306105495547348293749, −14.43344061342251990602688111731, −12.97664701560853149160404088547, −11.89293230051654397708049852819, −10.49157349351209535688193050598, −9.963629133540794207882785141408, −8.255622698155190429547567793677, −6.65752375968195805720507732929, −4.85098688399215011214238702043, −2.26623532713692530088546710173, 3.42987312905157534010236914006, 5.83768404749240717606545266475, 7.16251823162741675847358746285, 8.359819327199796891744287885900, 9.576821537056004788205838341040, 11.22716252147463454174169575444, 12.35110043118596423755352575705, 13.54999095046078507734294954532, 15.08039720005677721664988852131, 15.85212576093522570832154229519

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