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Properties

Label 2-775-31.25-c1-0-15
Degree $2$
Conductor $775$
Sign $0.739 - 0.673i$
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  + 0.869·2-s + (−0.0650 − 0.112i)3-s − 1.24·4-s + (−0.0565 − 0.0979i)6-s + (1.58 + 2.74i)7-s − 2.82·8-s + (1.49 − 2.58i)9-s + (0.267 − 0.463i)11-s + (0.0808 + 0.139i)12-s + (−0.667 + 1.15i)13-s + (1.38 + 2.39i)14-s + 0.0316·16-s + (3.19 + 5.53i)17-s + (1.29 − 2.24i)18-s + (2.14 + 3.71i)19-s + ⋯
L(s)  = 1  + 0.615·2-s + (−0.0375 − 0.0650i)3-s − 0.621·4-s + (−0.0230 − 0.0399i)6-s + (0.599 + 1.03i)7-s − 0.997·8-s + (0.497 − 0.861i)9-s + (0.0806 − 0.139i)11-s + (0.0233 + 0.0404i)12-s + (−0.185 + 0.320i)13-s + (0.368 + 0.639i)14-s + 0.00790·16-s + (0.774 + 1.34i)17-s + (0.305 − 0.529i)18-s + (0.491 + 0.851i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.739 - 0.673i$
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: \((2,\ 775,\ (\ :1/2),\ 0.739 - 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69065 + 0.654588i\)
\(L(\frac12)\) \(\approx\) \(1.69065 + 0.654588i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.22 - 3.62i)T \)
good2 \( 1 - 0.869T + 2T^{2} \)
3 \( 1 + (0.0650 + 0.112i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.58 - 2.74i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.267 + 0.463i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.667 - 1.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.19 - 5.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.14 - 3.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.35T + 23T^{2} \)
29 \( 1 - 1.35T + 29T^{2} \)
37 \( 1 + (0.0734 + 0.127i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.03 - 6.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.33 + 7.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.02T + 47T^{2} \)
53 \( 1 + (5.25 - 9.10i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.0144 + 0.0250i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + (-2.71 + 4.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.88 + 4.99i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.578 + 1.00i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.86 + 8.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.18 - 8.97i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−10.27668023027492916397993480038, −9.518109632646710988734101252517, −8.672677595846915048452586555388, −8.063805757954697697686042555580, −6.65648464451019506970471557582, −5.81380612367434982722510913682, −5.06787741987120546135734321275, −4.01441026238724647161939485292, −3.11555734132648676340538785210, −1.43932188705535151564884348690, 0.901866132746219977664412760550, 2.77331808861622549404487483542, 3.96277664878155674467627247305, 4.92081137290237215956499923541, 5.22605748414906124109792605655, 6.82004715264916833938453456380, 7.57339987066346042137351649573, 8.386991102243900902681581943178, 9.586192736102793775262218414084, 10.09501907291658188751648331987

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