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Properties

Label 2-507-13.12-c3-0-22
Degree $2$
Conductor $507$
Sign $-0.691 + 0.722i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
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  − 5.48i·2-s − 3·3-s − 22.0·4-s − 13.3i·5-s + 16.4i·6-s + 21.4i·7-s + 77.3i·8-s + 9·9-s − 73.0·10-s − 19.0i·11-s + 66.2·12-s + 117.·14-s + 39.9i·15-s + 247.·16-s + 71.7·17-s − 49.3i·18-s + ⋯
L(s)  = 1  − 1.93i·2-s − 0.577·3-s − 2.76·4-s − 1.19i·5-s + 1.11i·6-s + 1.15i·7-s + 3.41i·8-s + 0.333·9-s − 2.31·10-s − 0.522i·11-s + 1.59·12-s + 2.24·14-s + 0.687i·15-s + 3.86·16-s + 1.02·17-s − 0.646i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.159493803\)
\(L(\frac12)\) \(\approx\) \(1.159493803\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
13 \( 1 \)
good2 \( 1 + 5.48iT - 8T^{2} \)
5 \( 1 + 13.3iT - 125T^{2} \)
7 \( 1 - 21.4iT - 343T^{2} \)
11 \( 1 + 19.0iT - 1.33e3T^{2} \)
17 \( 1 - 71.7T + 4.91e3T^{2} \)
19 \( 1 - 102. iT - 6.85e3T^{2} \)
23 \( 1 - 37.8T + 1.21e4T^{2} \)
29 \( 1 - 40.8T + 2.43e4T^{2} \)
31 \( 1 - 6.05iT - 2.97e4T^{2} \)
37 \( 1 + 285. iT - 5.06e4T^{2} \)
41 \( 1 - 342. iT - 6.89e4T^{2} \)
43 \( 1 + 306.T + 7.95e4T^{2} \)
47 \( 1 - 346. iT - 1.03e5T^{2} \)
53 \( 1 - 398.T + 1.48e5T^{2} \)
59 \( 1 + 208. iT - 2.05e5T^{2} \)
61 \( 1 - 546.T + 2.26e5T^{2} \)
67 \( 1 - 678. iT - 3.00e5T^{2} \)
71 \( 1 + 957. iT - 3.57e5T^{2} \)
73 \( 1 - 270. iT - 3.89e5T^{2} \)
79 \( 1 + 1.03e3T + 4.93e5T^{2} \)
83 \( 1 - 1.06e3iT - 5.71e5T^{2} \)
89 \( 1 - 427. iT - 7.04e5T^{2} \)
97 \( 1 + 698. iT - 9.12e5T^{2} \)
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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

−10.23300798234047392883589008659, −9.537921687166493295444302054214, −8.719997490443314627514719857117, −8.102298983014769660503846655590, −5.74361215592091794934236023787, −5.26631231483737970053747564263, −4.21094508914097386177945053823, −3.05643467674154914568082796495, −1.70972719723493968409670822122, −0.77444573066904663437004541807, 0.65455162804122289254578698461, 3.46232334113034538413910219789, 4.54054655362193761648761548751, 5.44113841894727597478811296815, 6.61106651824709906441712742026, 7.02831071550964234390197393141, 7.57994670340263861461883982562, 8.760297661921181078615125806338, 10.04815665747183870923356866434, 10.34206626592644696198489780506

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