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Properties

Label 2-725-145.64-c1-0-35
Degree $2$
Conductor $725$
Sign $-0.642 + 0.765i$
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  + (1.57 − 1.97i)2-s + (0.0816 − 0.357i)3-s + (−0.971 − 4.25i)4-s + (−0.577 − 0.724i)6-s + (4.99 + 1.14i)7-s + (−5.37 − 2.59i)8-s + (2.58 + 1.24i)9-s + (−1.11 − 2.31i)11-s − 1.60·12-s + (−0.481 − 0.999i)13-s + (10.1 − 8.06i)14-s + (−5.70 + 2.74i)16-s − 1.36·17-s + (6.51 − 3.13i)18-s + (−1.06 + 0.242i)19-s + ⋯
L(s)  = 1  + (1.11 − 1.39i)2-s + (0.0471 − 0.206i)3-s + (−0.485 − 2.12i)4-s + (−0.235 − 0.295i)6-s + (1.88 + 0.431i)7-s + (−1.90 − 0.915i)8-s + (0.860 + 0.414i)9-s + (−0.336 − 0.698i)11-s − 0.462·12-s + (−0.133 − 0.277i)13-s + (2.70 − 2.15i)14-s + (−1.42 + 0.686i)16-s − 0.332·17-s + (1.53 − 0.739i)18-s + (−0.244 + 0.0556i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.642 + 0.765i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.642 + 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34145 - 2.87775i\)
\(L(\frac12)\) \(\approx\) \(1.34145 - 2.87775i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.04 - 1.87i)T \)
good2 \( 1 + (-1.57 + 1.97i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (-0.0816 + 0.357i)T + (-2.70 - 1.30i)T^{2} \)
7 \( 1 + (-4.99 - 1.14i)T + (6.30 + 3.03i)T^{2} \)
11 \( 1 + (1.11 + 2.31i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (0.481 + 0.999i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + 1.36T + 17T^{2} \)
19 \( 1 + (1.06 - 0.242i)T + (17.1 - 8.24i)T^{2} \)
23 \( 1 + (6.40 - 5.10i)T + (5.11 - 22.4i)T^{2} \)
31 \( 1 + (-5.93 - 4.73i)T + (6.89 + 30.2i)T^{2} \)
37 \( 1 + (2.74 + 1.32i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 - 7.19iT - 41T^{2} \)
43 \( 1 + (3.50 + 4.39i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (7.68 - 3.70i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (2.87 + 2.29i)T + (11.7 + 51.6i)T^{2} \)
59 \( 1 + 9.12T + 59T^{2} \)
61 \( 1 + (0.592 + 0.135i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (-0.977 + 2.03i)T + (-41.7 - 52.3i)T^{2} \)
71 \( 1 + (-8.05 + 3.87i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (1.86 + 2.33i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (-4.20 + 8.74i)T + (-49.2 - 61.7i)T^{2} \)
83 \( 1 + (-10.1 + 2.31i)T + (74.7 - 36.0i)T^{2} \)
89 \( 1 + (-4.10 - 3.27i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + (1.80 + 7.91i)T + (-87.3 + 42.0i)T^{2} \)
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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

−10.57795789374455524078192928071, −9.602182798602452633158968717130, −8.341047715075257231332825395255, −7.66749880174352967007679091540, −6.07557691377372354786905753009, −5.08010400355577808368826026547, −4.63516475825051646869112048347, −3.45954926443092301512752893780, −2.10811908541015692272592020685, −1.47723042887289977754842366710, 2.01315443394192672111298818199, 3.98114043340930919506109177337, 4.48225357587583541034632201914, 5.11102031323905808639676718800, 6.31137976635408715926214290359, 7.16598022858355708357526727531, 7.88509531624337634874212454167, 8.448391719904477566790767505040, 9.782366266068517311177844061227, 10.79419181654714632880537838496

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