Properties

Label 2-7e2-49.15-c3-0-3
Degree $2$
Conductor $49$
Sign $0.483 - 0.875i$
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  + (0.469 − 2.05i)2-s + (−3.54 + 4.44i)3-s + (3.19 + 1.53i)4-s + (−7.88 + 9.88i)5-s + (7.48 + 9.38i)6-s + (3.32 + 18.2i)7-s + (15.1 − 19.0i)8-s + (−1.19 − 5.22i)9-s + (16.6 + 20.8i)10-s + (2.22 − 9.75i)11-s + (−18.1 + 8.75i)12-s + (−2.85 + 12.5i)13-s + (39.0 + 1.71i)14-s + (−16.0 − 70.1i)15-s + (−14.3 − 18.0i)16-s + (45.2 − 21.7i)17-s + ⋯
L(s)  = 1  + (0.166 − 0.727i)2-s + (−0.682 + 0.855i)3-s + (0.399 + 0.192i)4-s + (−0.704 + 0.884i)5-s + (0.509 + 0.638i)6-s + (0.179 + 0.983i)7-s + (0.671 − 0.841i)8-s + (−0.0442 − 0.193i)9-s + (0.525 + 0.659i)10-s + (0.0610 − 0.267i)11-s + (−0.437 + 0.210i)12-s + (−0.0609 + 0.267i)13-s + (0.745 + 0.0328i)14-s + (−0.275 − 1.20i)15-s + (−0.224 − 0.281i)16-s + (0.645 − 0.310i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.483 - 0.875i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ 0.483 - 0.875i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.04418 + 0.616056i\)
\(L(\frac12)\) \(\approx\) \(1.04418 + 0.616056i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-3.32 - 18.2i)T \)
good2 \( 1 + (-0.469 + 2.05i)T + (-7.20 - 3.47i)T^{2} \)
3 \( 1 + (3.54 - 4.44i)T + (-6.00 - 26.3i)T^{2} \)
5 \( 1 + (7.88 - 9.88i)T + (-27.8 - 121. i)T^{2} \)
11 \( 1 + (-2.22 + 9.75i)T + (-1.19e3 - 577. i)T^{2} \)
13 \( 1 + (2.85 - 12.5i)T + (-1.97e3 - 953. i)T^{2} \)
17 \( 1 + (-45.2 + 21.7i)T + (3.06e3 - 3.84e3i)T^{2} \)
19 \( 1 + 42.4T + 6.85e3T^{2} \)
23 \( 1 + (45.8 + 22.1i)T + (7.58e3 + 9.51e3i)T^{2} \)
29 \( 1 + (-159. + 76.8i)T + (1.52e4 - 1.90e4i)T^{2} \)
31 \( 1 - 261.T + 2.97e4T^{2} \)
37 \( 1 + (-79.3 + 38.2i)T + (3.15e4 - 3.96e4i)T^{2} \)
41 \( 1 + (-254. + 319. i)T + (-1.53e4 - 6.71e4i)T^{2} \)
43 \( 1 + (-298. - 374. i)T + (-1.76e4 + 7.75e4i)T^{2} \)
47 \( 1 + (94.4 - 413. i)T + (-9.35e4 - 4.50e4i)T^{2} \)
53 \( 1 + (303. + 146. i)T + (9.28e4 + 1.16e5i)T^{2} \)
59 \( 1 + (400. + 502. i)T + (-4.57e4 + 2.00e5i)T^{2} \)
61 \( 1 + (-451. + 217. i)T + (1.41e5 - 1.77e5i)T^{2} \)
67 \( 1 + 1.02e3T + 3.00e5T^{2} \)
71 \( 1 + (-717. - 345. i)T + (2.23e5 + 2.79e5i)T^{2} \)
73 \( 1 + (-66.0 - 289. i)T + (-3.50e5 + 1.68e5i)T^{2} \)
79 \( 1 + 1.17e3T + 4.93e5T^{2} \)
83 \( 1 + (-113. - 495. i)T + (-5.15e5 + 2.48e5i)T^{2} \)
89 \( 1 + (199. + 874. i)T + (-6.35e5 + 3.05e5i)T^{2} \)
97 \( 1 - 185.T + 9.12e5T^{2} \)
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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

−15.64713967892351104504638838933, −14.32110986195333076965755900889, −12.46620983355619601807665211920, −11.53135151268915634357952637661, −10.94651391809547643438763899785, −9.825733095152971138711813427616, −7.86993129600860938280241207460, −6.22416168386628717640934170208, −4.38096518911325531139065936050, −2.79635163284564371020746507018, 1.04843766827590131206218842948, 4.56087610838537095695047722596, 6.11031925334697072922146877360, 7.27201967728604770859114887187, 8.174825224358584520691763255177, 10.38095654406606833425509550009, 11.68590818004761219543523698503, 12.52616815029047372678856212903, 13.79055529747615677732718700838, 15.07158209457717223918203074057

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