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Properties

Label 2-725-145.64-c1-0-24
Degree $2$
Conductor $725$
Sign $0.857 + 0.514i$
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  + (−0.791 + 0.992i)2-s + (0.583 − 2.55i)3-s + (0.0867 + 0.380i)4-s + (2.07 + 2.60i)6-s + (−0.00231 − 0.000528i)7-s + (−2.73 − 1.31i)8-s + (−3.48 − 1.67i)9-s + (1.31 + 2.73i)11-s + 1.02·12-s + (−0.518 − 1.07i)13-s + (0.00235 − 0.00188i)14-s + (2.76 − 1.33i)16-s + 4.90·17-s + (4.42 − 2.13i)18-s + (5.21 − 1.18i)19-s + ⋯
L(s)  = 1  + (−0.559 + 0.701i)2-s + (0.336 − 1.47i)3-s + (0.0433 + 0.190i)4-s + (0.846 + 1.06i)6-s + (−0.000875 − 0.000199i)7-s + (−0.965 − 0.465i)8-s + (−1.16 − 0.559i)9-s + (0.396 + 0.823i)11-s + 0.295·12-s + (−0.143 − 0.298i)13-s + (0.000630 − 0.000502i)14-s + (0.691 − 0.332i)16-s + 1.18·17-s + (1.04 − 0.502i)18-s + (1.19 − 0.272i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.857 + 0.514i$
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: \((2,\ 725,\ (\ :1/2),\ 0.857 + 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23272 - 0.341290i\)
\(L(\frac12)\) \(\approx\) \(1.23272 - 0.341290i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 + 4.92i)T \)
good2 \( 1 + (0.791 - 0.992i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (-0.583 + 2.55i)T + (-2.70 - 1.30i)T^{2} \)
7 \( 1 + (0.00231 + 0.000528i)T + (6.30 + 3.03i)T^{2} \)
11 \( 1 + (-1.31 - 2.73i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (0.518 + 1.07i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 - 4.90T + 17T^{2} \)
19 \( 1 + (-5.21 + 1.18i)T + (17.1 - 8.24i)T^{2} \)
23 \( 1 + (-5.01 + 3.99i)T + (5.11 - 22.4i)T^{2} \)
31 \( 1 + (6.31 + 5.03i)T + (6.89 + 30.2i)T^{2} \)
37 \( 1 + (1.37 + 0.664i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 - 8.22iT - 41T^{2} \)
43 \( 1 + (3.45 + 4.33i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (0.757 - 0.364i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (0.783 + 0.624i)T + (11.7 + 51.6i)T^{2} \)
59 \( 1 - 15.1T + 59T^{2} \)
61 \( 1 + (1.58 + 0.362i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (-3.19 + 6.63i)T + (-41.7 - 52.3i)T^{2} \)
71 \( 1 + (-8.87 + 4.27i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-4.08 - 5.12i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (-0.570 + 1.18i)T + (-49.2 - 61.7i)T^{2} \)
83 \( 1 + (12.8 - 2.92i)T + (74.7 - 36.0i)T^{2} \)
89 \( 1 + (-10.4 - 8.36i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + (-1.63 - 7.15i)T + (-87.3 + 42.0i)T^{2} \)
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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

−9.940834737592553198612806211604, −9.253586250027513591040036772551, −8.194438924694410713369648306759, −7.71622346544002347950652391642, −7.01607357825525431745945747835, −6.41806910838487212955090239325, −5.23623420669250006312320614070, −3.49949221360752340939483215971, −2.40532941950601116457939394649, −0.905791733054522695147570402537, 1.31402329830416574001201392912, 3.12425126368917374663851328249, 3.53678188276601610406702185554, 5.11318941519435143719247447270, 5.60718128542493806491302534576, 7.08185782217265324646040510615, 8.446963919758279607714501073265, 9.092523515876308122558669968128, 9.711388800257700049214250009027, 10.30162262407987500458109452642

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