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Properties

Label 2-925-37.10-c1-0-29
Degree $2$
Conductor $925$
Sign $0.957 + 0.289i$
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  + (0.5 − 0.866i)2-s + (0.500 + 0.866i)4-s + (1 + 1.73i)7-s + 3·8-s + (1.5 − 2.59i)9-s − 2·11-s + (−1 − 1.73i)13-s + 1.99·14-s + (0.500 − 0.866i)16-s + (1.5 − 2.59i)17-s + (−1.5 − 2.59i)18-s + (3 + 5.19i)19-s + (−1 + 1.73i)22-s + 4·23-s − 1.99·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + (0.377 + 0.654i)7-s + 1.06·8-s + (0.5 − 0.866i)9-s − 0.603·11-s + (−0.277 − 0.480i)13-s + 0.534·14-s + (0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (−0.353 − 0.612i)18-s + (0.688 + 1.19i)19-s + (−0.213 + 0.369i)22-s + 0.834·23-s − 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.957 + 0.289i$
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: \((2,\ 925,\ (\ :1/2),\ 0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31522 - 0.343053i\)
\(L(\frac12)\) \(\approx\) \(2.31522 - 0.343053i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.5 - 2.59i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−10.10015082217828999944431246530, −9.395327023802856567894428135612, −8.212008649394670232277399050386, −7.62567501985610028733689108645, −6.66415431427463290965848929672, −5.48142445831349732251106790332, −4.63251522624793846765833196404, −3.43064357335669774246762756604, −2.73057961931775557151792166753, −1.36473062318667112130856009604, 1.26217481548301620448055469066, 2.53157389423715846901762624518, 4.20031771808600927924622335608, 4.93956128784287728518583243545, 5.63402476117309804912883727040, 6.94380142280896450913929201412, 7.30231305795038479880177906080, 8.134005279940897243835588009731, 9.336864264650541008897033432223, 10.33112038729168263640636751165

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