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Properties

Label 2-725-145.4-c1-0-6
Degree $2$
Conductor $725$
Sign $0.334 - 0.942i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.334 - 0.942i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.334 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.439046 + 0.309983i\)
\(L(\frac12)\) \(\approx\) \(0.439046 + 0.309983i\)
\(L(\frac{3}{2})\) 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

−10.74630233168970879575569673229, −10.05896555093049052863870074491, −9.279340487603795387007197143970, −7.960113202338568561721728459178, −7.19104739142773554805846962389, −5.73741555696102780209373526242, −4.90435505403278090374865023373, −4.13743372576926498138738931235, −2.63771592696865771942770820471, −1.99801178552299970557669525765, 0.24483049197042489915897939440, 2.51979256588982308737246485987, 4.03164703651201301195147964144, 5.37591514567403233880574207618, 5.66031950704993095542732477671, 6.66395595428834368340098286352, 7.45151682916940617352209407362, 8.250149330313871931373366584355, 8.992369873332694600059585058347, 10.45228364597503181191242662686

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