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Properties

Label 2-725-145.57-c0-0-0
Degree $2$
Conductor $725$
Sign $0.525 - 0.850i$
Analytic cond. $0.361822$
Root an. cond. $0.601516$
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  + i·4-s + (1 + i)7-s − i·9-s + (−1 + i)13-s − 16-s + (1 − i)23-s + (−1 + i)28-s + i·29-s + 36-s + i·49-s + (−1 − i)52-s + (1 − i)53-s − 2i·59-s + (1 − i)63-s − i·64-s + ⋯
L(s)  = 1  + i·4-s + (1 + i)7-s − i·9-s + (−1 + i)13-s − 16-s + (1 − i)23-s + (−1 + i)28-s + i·29-s + 36-s + i·49-s + (−1 − i)52-s + (1 − i)53-s − 2i·59-s + (1 − i)63-s − i·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(0.361822\)
Root analytic conductor: \(0.601516\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001565417\)
\(L(\frac12)\) \(\approx\) \(1.001565417\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 - iT \)
good2 \( 1 - iT^{2} \)
3 \( 1 + iT^{2} \)
7 \( 1 + (-1 - i)T + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−11.03587404064170320549590750143, −9.623204768969270093328618386830, −8.881002273048862580135578209051, −8.368381380136127682650206194619, −7.23423471355048718072329042542, −6.56465872708139332683908208791, −5.17543925260232201009803150902, −4.38227304064454782608807153750, −3.13317389367264562143096942095, −2.05841030962527830477507538316, 1.25422002409433749954878840117, 2.56346500264941068433264239817, 4.32330256817179189324365120004, 5.05724915634242640488193150188, 5.77418450959617869697732827195, 7.29137821434148289950849904461, 7.61055636225506915729162095210, 8.764962612967294719663789221999, 9.932294100703347519451800294365, 10.45190520524103108025931289373

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