Properties

Label 2-7e2-7.6-c6-0-5
Degree $2$
Conductor $49$
Sign $-0.755 - 0.654i$
Analytic cond. $11.2726$
Root an. cond. $3.35747$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.24·2-s + 26.6i·3-s + 3.94·4-s + 79.1i·5-s + 219. i·6-s − 495.·8-s + 18.9·9-s + 652. i·10-s − 1.70e3·11-s + 105. i·12-s + 3.12e3i·13-s − 2.10e3·15-s − 4.33e3·16-s − 4.07e3i·17-s + 155.·18-s + 5.87e3i·19-s + ⋯
L(s)  = 1  + 1.03·2-s + 0.986i·3-s + 0.0615·4-s + 0.633i·5-s + 1.01i·6-s − 0.966·8-s + 0.0259·9-s + 0.652i·10-s − 1.28·11-s + 0.0607i·12-s + 1.42i·13-s − 0.625·15-s − 1.05·16-s − 0.829i·17-s + 0.0267·18-s + 0.856i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.755 - 0.654i$
Analytic conductor: \(11.2726\)
Root analytic conductor: \(3.35747\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3),\ -0.755 - 0.654i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.700761 + 1.87960i\)
\(L(\frac12)\) \(\approx\) \(0.700761 + 1.87960i\)
\(L(4)\) 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 \( 1 - 8.24T + 64T^{2} \)
3 \( 1 - 26.6iT - 729T^{2} \)
5 \( 1 - 79.1iT - 1.56e4T^{2} \)
11 \( 1 + 1.70e3T + 1.77e6T^{2} \)
13 \( 1 - 3.12e3iT - 4.82e6T^{2} \)
17 \( 1 + 4.07e3iT - 2.41e7T^{2} \)
19 \( 1 - 5.87e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.33e4T + 1.48e8T^{2} \)
29 \( 1 - 6.51e3T + 5.94e8T^{2} \)
31 \( 1 - 1.19e4iT - 8.87e8T^{2} \)
37 \( 1 + 4.64e3T + 2.56e9T^{2} \)
41 \( 1 - 1.93e4iT - 4.75e9T^{2} \)
43 \( 1 - 9.16e4T + 6.32e9T^{2} \)
47 \( 1 + 6.44e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.49e5T + 2.21e10T^{2} \)
59 \( 1 + 6.10e4iT - 4.21e10T^{2} \)
61 \( 1 + 9.86e4iT - 5.15e10T^{2} \)
67 \( 1 + 3.11e5T + 9.04e10T^{2} \)
71 \( 1 + 4.01e5T + 1.28e11T^{2} \)
73 \( 1 - 6.72e5iT - 1.51e11T^{2} \)
79 \( 1 + 3.20e5T + 2.43e11T^{2} \)
83 \( 1 - 8.32e5iT - 3.26e11T^{2} \)
89 \( 1 - 3.79e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.05e6iT - 8.32e11T^{2} \)
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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

−14.70508451855592270398692935937, −13.85255041457500617871404413898, −12.67397310798516072361735771466, −11.31338766131057386837370576499, −10.15904123487134995532319521680, −8.986511393728475466669482266752, −6.98312834902249422565613045275, −5.31010751931213115390759913299, −4.27805006486738202632136614221, −2.91286435839154014186397358143, 0.65181894478424413101430559891, 2.81081303314465135137337813311, 4.75009260802143116111713056445, 5.86857368351518607262357309478, 7.50671660623642406040306689109, 8.749214729272753463498394680270, 10.56061104678245225418429192911, 12.25797136134747927642360109979, 13.12773119022028595973510813373, 13.22702170543242664228456293416

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