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Properties

Label 2-725-145.109-c1-0-39
Degree $2$
Conductor $725$
Sign $-0.954 - 0.297i$
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  + (−0.416 − 1.82i)2-s + (0.896 + 0.431i)3-s + (−1.34 + 0.649i)4-s + (0.413 − 1.81i)6-s + (−0.393 + 0.817i)7-s + (−0.587 − 0.736i)8-s + (−1.25 − 1.57i)9-s + (−4.84 − 3.85i)11-s − 1.48·12-s + (1.31 + 1.04i)13-s + (1.65 + 0.377i)14-s + (−2.96 + 3.71i)16-s + 2.61·17-s + (−2.34 + 2.93i)18-s + (−0.943 − 1.95i)19-s + ⋯
L(s)  = 1  + (−0.294 − 1.28i)2-s + (0.517 + 0.249i)3-s + (−0.673 + 0.324i)4-s + (0.169 − 0.740i)6-s + (−0.148 + 0.308i)7-s + (−0.207 − 0.260i)8-s + (−0.417 − 0.523i)9-s + (−1.45 − 1.16i)11-s − 0.429·12-s + (0.364 + 0.290i)13-s + (0.441 + 0.100i)14-s + (−0.741 + 0.929i)16-s + 0.633·17-s + (−0.552 + 0.692i)18-s + (−0.216 − 0.449i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.954 - 0.297i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.954 - 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.131779 + 0.865379i\)
\(L(\frac12)\) \(\approx\) \(0.131779 + 0.865379i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.35 + 4.21i)T \)
good2 \( 1 + (0.416 + 1.82i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-0.896 - 0.431i)T + (1.87 + 2.34i)T^{2} \)
7 \( 1 + (0.393 - 0.817i)T + (-4.36 - 5.47i)T^{2} \)
11 \( 1 + (4.84 + 3.85i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.31 - 1.04i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 - 2.61T + 17T^{2} \)
19 \( 1 + (0.943 + 1.95i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (-2.64 - 0.604i)T + (20.7 + 9.97i)T^{2} \)
31 \( 1 + (6.14 - 1.40i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (5.71 + 7.16i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 9.46iT - 41T^{2} \)
43 \( 1 + (1.36 - 5.98i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (3.06 - 3.84i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (4.97 - 1.13i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 - 9.00T + 59T^{2} \)
61 \( 1 + (-0.934 + 1.93i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (-8.46 + 6.75i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (-2.92 + 3.66i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-0.611 + 2.67i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (4.33 - 3.45i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-0.683 - 1.41i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (-7.82 + 1.78i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-14.9 + 7.21i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.00897608788466487229727324572, −9.121343169399655913692702267798, −8.687667372807774427075835195721, −7.62751580821645130313038665508, −6.22851817236418914519414742663, −5.35601741651583894299351569878, −3.74068787371391697112998225987, −3.12738078813204344574690306934, −2.20490752529819689592265442985, −0.43247575087600326267661357202, 2.12335668347570756145293104227, 3.30346531258007165298429760159, 5.02875865138348719934296460729, 5.50285684733061049082781778669, 6.82352546533808266020221959357, 7.46560560508662368919494133044, 8.088143302933411852671149354881, 8.734613217380343273729407840791, 9.870410415003507761668221817277, 10.59760441885518282116876318641

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