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Properties

Label 2-925-37.10-c1-0-51
Degree $2$
Conductor $925$
Sign $-0.964 + 0.264i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
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  + (1.24 − 2.14i)2-s + (1.10 + 1.90i)3-s + (−2.07 − 3.59i)4-s + 5.46·6-s + (−1.91 − 3.31i)7-s − 5.35·8-s + (−0.929 + 1.60i)9-s − 4.82·11-s + (4.58 − 7.93i)12-s + (−2.50 − 4.34i)13-s − 9.49·14-s + (−2.48 + 4.29i)16-s + (2.38 − 4.13i)17-s + (2.30 + 3.99i)18-s + (2.59 + 4.49i)19-s + ⋯
L(s)  = 1  + (0.877 − 1.51i)2-s + (0.636 + 1.10i)3-s + (−1.03 − 1.79i)4-s + 2.23·6-s + (−0.723 − 1.25i)7-s − 1.89·8-s + (−0.309 + 0.536i)9-s − 1.45·11-s + (1.32 − 2.29i)12-s + (−0.695 − 1.20i)13-s − 2.53·14-s + (−0.620 + 1.07i)16-s + (0.579 − 1.00i)17-s + (0.543 + 0.941i)18-s + (0.595 + 1.03i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.964 + 0.264i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.964 + 0.264i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.283581 - 2.10695i\)
\(L(\frac12)\) \(\approx\) \(0.283581 - 2.10695i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 + (6.03 - 0.731i)T \)
good2 \( 1 + (-1.24 + 2.14i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.10 - 1.90i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.91 + 3.31i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + (2.50 + 4.34i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.38 + 4.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.59 - 4.49i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.630T + 23T^{2} \)
29 \( 1 - 4.07T + 29T^{2} \)
31 \( 1 - 1.59T + 31T^{2} \)
41 \( 1 + (0.00413 + 0.00716i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 - 2.48T + 47T^{2} \)
53 \( 1 + (0.157 - 0.272i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.51 + 13.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.80 + 3.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.62 - 8.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.40 - 4.15i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.84T + 73T^{2} \)
79 \( 1 + (1.43 + 2.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.38 - 4.12i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.93 - 15.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.19T + 97T^{2} \)
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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

−10.09135037409070618718995286855, −9.596709342892845665709887274955, −8.170500202229103955802874494766, −7.26218095175639073935479519050, −5.54534415201309410395756247564, −4.92493020639855102795618748644, −3.96525053981681696320495022151, −3.20488499888455604344533579004, −2.69233571115205953028567187562, −0.67634872667704018663428806762, 2.29519827201390490324014667250, 3.09077598004534223620470267841, 4.58512389361581909586535017523, 5.50680905981915057079765270673, 6.22519028136281566596222699041, 7.10968454665668633165643758675, 7.58881408005103982568577674741, 8.518918246601752857119003218414, 9.009537282653473555798823507354, 10.24731516409644267469828669441

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