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Properties

Label 2-725-145.109-c1-0-9
Degree $2$
Conductor $725$
Sign $0.435 + 0.900i$
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  + (−0.217 − 0.951i)2-s + (−2.24 − 1.07i)3-s + (0.943 − 0.454i)4-s + (−0.540 + 2.36i)6-s + (−0.892 + 1.85i)7-s + (−1.85 − 2.32i)8-s + (1.98 + 2.49i)9-s + (4.43 + 3.53i)11-s − 2.60·12-s + (4.41 + 3.52i)13-s + (1.95 + 0.446i)14-s + (−0.504 + 0.632i)16-s + 2.55·17-s + (1.94 − 2.43i)18-s + (−2.69 − 5.59i)19-s + ⋯
L(s)  = 1  + (−0.153 − 0.672i)2-s + (−1.29 − 0.623i)3-s + (0.471 − 0.227i)4-s + (−0.220 + 0.966i)6-s + (−0.337 + 0.700i)7-s + (−0.655 − 0.822i)8-s + (0.662 + 0.831i)9-s + (1.33 + 1.06i)11-s − 0.752·12-s + (1.22 + 0.977i)13-s + (0.523 + 0.119i)14-s + (−0.126 + 0.158i)16-s + 0.619·17-s + (0.457 − 0.573i)18-s + (−0.617 − 1.28i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.435 + 0.900i$
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: \((2,\ 725,\ (\ :1/2),\ 0.435 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939215 - 0.589086i\)
\(L(\frac12)\) \(\approx\) \(0.939215 - 0.589086i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.783 - 5.32i)T \)
good2 \( 1 + (0.217 + 0.951i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (2.24 + 1.07i)T + (1.87 + 2.34i)T^{2} \)
7 \( 1 + (0.892 - 1.85i)T + (-4.36 - 5.47i)T^{2} \)
11 \( 1 + (-4.43 - 3.53i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-4.41 - 3.52i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 - 2.55T + 17T^{2} \)
19 \( 1 + (2.69 + 5.59i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (-4.01 - 0.916i)T + (20.7 + 9.97i)T^{2} \)
31 \( 1 + (-0.0148 + 0.00339i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (-0.969 - 1.21i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 - 2.39iT - 41T^{2} \)
43 \( 1 + (-1.96 + 8.61i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-1.27 + 1.59i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.219 + 0.0501i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + (-3.87 + 8.03i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (-11.6 + 9.27i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (-0.471 + 0.590i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (0.334 - 1.46i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (-6.28 + 5.01i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-1.53 - 3.19i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (4.73 - 1.07i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-3.31 + 1.59i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.63301191881175421719848363525, −9.369407253214263704043084656891, −8.984813666511696568622549608888, −7.13200935326264443450493504481, −6.61783294019687780201380055305, −6.08665983221590447809340841349, −4.91746571242311235789647407224, −3.56206353476915314568785724440, −2.01738364218788531648469565846, −1.08092050430464018882230933613, 0.954430278163929561071398977224, 3.32869167271972065507698903114, 4.12100082925015918973271751631, 5.71622181092813631513946841776, 5.99435675616007129399359412826, 6.73772605512896595276984315856, 7.917383216575781978970220904328, 8.681288740576149667100443941854, 9.874851252560084139896064775691, 10.78059662348971621440238936389

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