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Properties

Label 2-7e2-7.4-c3-0-4
Degree $2$
Conductor $49$
Sign $0.947 - 0.318i$
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  + (1.76 + 3.05i)2-s + (3.91 − 6.77i)3-s + (−2.23 + 3.87i)4-s + (1.03 + 1.79i)5-s + 27.6·6-s + 12.4·8-s + (−17.0 − 29.6i)9-s + (−3.66 + 6.35i)10-s + (−24.5 + 42.5i)11-s + (17.4 + 30.2i)12-s − 44.8·13-s + 16.2·15-s + (39.8 + 69.0i)16-s + (−13.2 + 22.9i)17-s + (60.3 − 104. i)18-s + (−38.8 − 67.3i)19-s + ⋯
L(s)  = 1  + (0.624 + 1.08i)2-s + (0.752 − 1.30i)3-s + (−0.279 + 0.483i)4-s + (0.0928 + 0.160i)5-s + 1.87·6-s + 0.551·8-s + (−0.633 − 1.09i)9-s + (−0.115 + 0.200i)10-s + (−0.674 + 1.16i)11-s + (0.420 + 0.728i)12-s − 0.956·13-s + 0.279·15-s + (0.623 + 1.07i)16-s + (−0.189 + 0.327i)17-s + (0.790 − 1.36i)18-s + (−0.469 − 0.812i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ 0.947 - 0.318i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.19645 + 0.359221i\)
\(L(\frac12)\) \(\approx\) \(2.19645 + 0.359221i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.76 - 3.05i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-3.91 + 6.77i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-1.03 - 1.79i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (24.5 - 42.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 44.8T + 2.19e3T^{2} \)
17 \( 1 + (13.2 - 22.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (38.8 + 67.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (27.8 + 48.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 121.T + 2.43e4T^{2} \)
31 \( 1 + (152. - 264. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (38.5 + 66.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 248.T + 6.89e4T^{2} \)
43 \( 1 + 147.T + 7.95e4T^{2} \)
47 \( 1 + (134. + 233. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-70.5 + 122. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-212. + 367. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-293. - 509. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-89.8 + 155. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 674.T + 3.57e5T^{2} \)
73 \( 1 + (118. - 205. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (247. + 429. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 24.4T + 5.71e5T^{2} \)
89 \( 1 + (536. + 928. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.66e3T + 9.12e5T^{2} \)
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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

−14.80626463934405994392326834585, −14.25343209891883154152047173061, −13.06311679312541133344475229419, −12.44083374549411526555415168260, −10.33956291321829620853877192189, −8.451912762127971553321840867752, −7.30406778523848428895738047012, −6.67424031866734028450471019162, −4.86072411061712537847113237302, −2.25176300575461817891705979816, 2.69872940460800571062035467619, 3.90370963521978325696160293555, 5.21526535009488464766798305784, 7.979329703122161805662615569252, 9.429229473283096612614193429971, 10.41725571321107311986196552789, 11.37375924458629012659605838061, 12.80639316877543168650461741275, 13.87251789219234183894277692301, 14.80685271922886766576198392494

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