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Properties

Label 1-7e2-49.29-r0-0-0
Degree $1$
Conductor $49$
Sign $-0.672 + 0.740i$
Analytic cond. $0.227555$
Root an. cond. $0.227555$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 + 0.781i)2-s + (−0.900 + 0.433i)3-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s + (−0.900 − 0.433i)6-s + (−0.900 + 0.433i)8-s + (0.623 − 0.781i)9-s + (−0.900 − 0.433i)10-s + (0.623 + 0.781i)11-s + (−0.222 − 0.974i)12-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + (−0.222 − 0.974i)17-s + 18-s + 19-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (−0.900 + 0.433i)3-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s + (−0.900 − 0.433i)6-s + (−0.900 + 0.433i)8-s + (0.623 − 0.781i)9-s + (−0.900 − 0.433i)10-s + (0.623 + 0.781i)11-s + (−0.222 − 0.974i)12-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + (−0.222 − 0.974i)17-s + 18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.672 + 0.740i$
Analytic conductor: \(0.227555\)
Root analytic conductor: \(0.227555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (0:\ ),\ -0.672 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3084016739 + 0.6966847535i\)
\(L(\frac12)\) \(\approx\) \(0.3084016739 + 0.6966847535i\)
\(L(1)\) \(\approx\) \(0.6673565200 + 0.6279864890i\)
\(L(1)\) \(\approx\) \(0.6673565200 + 0.6279864890i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 + (-0.900 - 0.433i)T \)
61 \( 1 + (-0.222 - 0.974i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−33.11484175795562833495853478095, −32.236018278314702227574836393, −30.79743730706022787391175372554, −30.11667416769505129577758486392, −28.8155034193021144430942922568, −27.99680404912355304610707158426, −27.061499162392013220642195518484, −24.621526593735992455521018558704, −23.88846337712835952973659075678, −22.822612007233742848441980031426, −21.94942948652353453437163745056, −20.40135811763955304356491623265, −19.3157260086256144632086963924, −18.2515969252552190770872343907, −16.602729950860218375770954336947, −15.336150675731618719359333403939, −13.58282255614661952312209072955, −12.4345250357034266273340176781, −11.53897447740747797473114158446, −10.488999912389959454635374465411, −8.42221009240629092763690514606, −6.43699204361261560893900551562, −5.07778570447666233172176451792, −3.59021991559673552754788675845, −1.10243920634589678238791755155, 3.67103236213400625148587958008, 4.78077931061807683666236754315, 6.41813870650314258742355562036, 7.46416823385243877355683560175, 9.40592740050215438917767077792, 11.43939016912632378456591074570, 12.06750049661719219145538821757, 13.9030365617759646104405128266, 15.35285302914697572706954057299, 16.021899096444041578191448182, 17.299186926816323441417437123668, 18.486531230616380999517181919975, 20.43997556185653167777783322780, 21.85059889447489981101086826274, 22.8358676312552065233982276615, 23.43547513203816601438432100477, 24.73017093373225737163692720289, 26.30410179679758554246256323106, 27.14337686086795957218035752676, 28.32694216474670484248322154627, 29.95184959307023862219221017367, 30.97347201406057966729790640252, 32.0956471197246953989057623712, 33.4980867210934649201770941163, 33.84860728268130797567468716364

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