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Properties

Label 2-775-31.25-c1-0-39
Degree $2$
Conductor $775$
Sign $0.894 - 0.446i$
Analytic cond. $6.18840$
Root an. cond. $2.48765$
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  + 2.60·2-s + (0.801 + 1.38i)3-s + 4.77·4-s + (2.08 + 3.61i)6-s + (−1.16 − 2.01i)7-s + 7.22·8-s + (0.214 − 0.372i)9-s + (0.138 − 0.240i)11-s + (3.82 + 6.63i)12-s + (−1.66 + 2.88i)13-s + (−3.03 − 5.25i)14-s + 9.26·16-s + (2.65 + 4.59i)17-s + (0.559 − 0.969i)18-s + (−2.75 − 4.76i)19-s + ⋯
L(s)  = 1  + 1.84·2-s + (0.462 + 0.801i)3-s + 2.38·4-s + (0.851 + 1.47i)6-s + (−0.440 − 0.763i)7-s + 2.55·8-s + (0.0716 − 0.124i)9-s + (0.0418 − 0.0724i)11-s + (1.10 + 1.91i)12-s + (−0.461 + 0.798i)13-s + (−0.810 − 1.40i)14-s + 2.31·16-s + (0.644 + 1.11i)17-s + (0.131 − 0.228i)18-s + (−0.631 − 1.09i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.894 - 0.446i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.894 - 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.85428 + 1.14401i\)
\(L(\frac12)\) \(\approx\) \(4.85428 + 1.14401i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 + (-3.07 - 4.64i)T \)
good2 \( 1 - 2.60T + 2T^{2} \)
3 \( 1 + (-0.801 - 1.38i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.16 + 2.01i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.138 + 0.240i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.66 - 2.88i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.65 - 4.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.75 + 4.76i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.94T + 23T^{2} \)
29 \( 1 + 8.94T + 29T^{2} \)
37 \( 1 + (0.483 + 0.837i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.00 + 6.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.09 - 1.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + (-3.92 + 6.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.20 - 9.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.839T + 61T^{2} \)
67 \( 1 + (2.69 - 4.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.605 - 1.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.45 - 5.97i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.17 - 2.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.39 + 12.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.21T + 89T^{2} \)
97 \( 1 - 4.65T + 97T^{2} \)
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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

−10.45454100550326392340298055226, −9.864331354433374365825893622642, −8.692421856023290480722716729712, −7.36159214814131094925289436820, −6.65263367957097479988633332543, −5.76752801980432256903049493307, −4.59505107656538310897444932508, −3.97403890089309230128640135118, −3.39041612300407195974924767893, −2.06440801125857399634958688614, 1.95511041745181621423491319587, 2.71595564619731646797837945979, 3.70118677595344625503069470906, 4.89882917094738505026237323350, 5.78855083034932154868363021674, 6.41314006303693630469962974536, 7.58677448681530173144104268918, 7.956974108042432970383357008709, 9.540865234364616149815438065932, 10.45958870592403665036692876039

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