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Properties

Label 2-725-145.133-c1-0-24
Degree $2$
Conductor $725$
Sign $-0.939 - 0.341i$
Analytic cond. $5.78915$
Root an. cond. $2.40606$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.32i·2-s − 3.11·3-s − 3.39·4-s + 7.22i·6-s + (2.50 − 2.50i)7-s + 3.23i·8-s + 6.68·9-s + (2.71 − 2.71i)11-s + 10.5·12-s + (2.88 − 2.88i)13-s + (−5.81 − 5.81i)14-s + 0.734·16-s − 4.08i·17-s − 15.5i·18-s + (3.71 + 3.71i)19-s + ⋯
L(s)  = 1  − 1.64i·2-s − 1.79·3-s − 1.69·4-s + 2.95i·6-s + (0.945 − 0.945i)7-s + 1.14i·8-s + 2.22·9-s + (0.817 − 0.817i)11-s + 3.05·12-s + (0.798 − 0.798i)13-s + (−1.55 − 1.55i)14-s + 0.183·16-s − 0.991i·17-s − 3.66i·18-s + (0.851 + 0.851i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.939 - 0.341i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.939 - 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160255 + 0.910282i\)
\(L(\frac12)\) \(\approx\) \(0.160255 + 0.910282i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.51 + 4.07i)T \)
good2 \( 1 + 2.32iT - 2T^{2} \)
3 \( 1 + 3.11T + 3T^{2} \)
7 \( 1 + (-2.50 + 2.50i)T - 7iT^{2} \)
11 \( 1 + (-2.71 + 2.71i)T - 11iT^{2} \)
13 \( 1 + (-2.88 + 2.88i)T - 13iT^{2} \)
17 \( 1 + 4.08iT - 17T^{2} \)
19 \( 1 + (-3.71 - 3.71i)T + 19iT^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
31 \( 1 + (-1.31 + 1.31i)T - 31iT^{2} \)
37 \( 1 - 7.39T + 37T^{2} \)
41 \( 1 + (1.29 + 1.29i)T + 41iT^{2} \)
43 \( 1 + 6.79T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + (-3.85 - 3.85i)T + 53iT^{2} \)
59 \( 1 + 0.511iT - 59T^{2} \)
61 \( 1 + (-4.83 + 4.83i)T - 61iT^{2} \)
67 \( 1 + (-5.02 - 5.02i)T + 67iT^{2} \)
71 \( 1 - 6.77iT - 71T^{2} \)
73 \( 1 - 5.31iT - 73T^{2} \)
79 \( 1 + (8.66 + 8.66i)T + 79iT^{2} \)
83 \( 1 + (8.31 + 8.31i)T + 83iT^{2} \)
89 \( 1 + (4.54 + 4.54i)T + 89iT^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−10.20823812997379785265924128451, −9.773737316153786317473637564594, −8.363770627148565789752562472010, −7.21943726552328666623301938951, −6.08910496797383430868775060384, −5.13179573203666084575249409245, −4.31230878001516184282380482180, −3.36733718386852788580156100215, −1.30535133733217813200189201215, −0.797908907829593715444239254344, 1.47034089576459317219466999155, 4.34049948599834427583192613746, 4.94153785646360319090891099089, 5.63528604408041664788519143900, 6.66453337542988402591095728223, 6.76598339062572466476955235970, 8.124917966005659538497544067382, 8.920091058327962105248084705100, 9.873011368083252466584519849674, 11.18088986614711944645040563338

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