↑

Properties

Label 2-275-55.7-c1-0-2
Degree $2$
Conductor $275$
Sign $0.363 - 0.931i$
Analytic cond. $2.19588$
Root an. cond. $1.48185$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.50 + 0.237i)2-s + (0.361 + 0.710i)3-s + (0.295 − 0.0959i)4-s + (−0.712 − 0.980i)6-s + (−0.170 − 0.0869i)7-s + (2.28 − 1.16i)8-s + (1.38 − 1.91i)9-s + (1.77 + 2.80i)11-s + (0.175 + 0.175i)12-s + (0.484 + 3.05i)13-s + (0.276 + 0.0899i)14-s + (−3.66 + 2.65i)16-s + (−0.579 + 3.66i)17-s + (−1.63 + 3.20i)18-s + (−0.229 + 0.707i)19-s + ⋯
L(s)  = 1  + (−1.06 + 0.168i)2-s + (0.208 + 0.410i)3-s + (0.147 − 0.0479i)4-s + (−0.290 − 0.400i)6-s + (−0.0644 − 0.0328i)7-s + (0.808 − 0.412i)8-s + (0.463 − 0.637i)9-s + (0.535 + 0.844i)11-s + (0.0505 + 0.0505i)12-s + (0.134 + 0.847i)13-s + (0.0739 + 0.0240i)14-s + (−0.915 + 0.664i)16-s + (−0.140 + 0.887i)17-s + (−0.384 + 0.754i)18-s + (−0.0527 + 0.162i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.363 - 0.931i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.363 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.632990 + 0.432562i\)
\(L(\frac12)\) \(\approx\) \(0.632990 + 0.432562i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + (-1.77 - 2.80i)T \)
good2 \( 1 + (1.50 - 0.237i)T + (1.90 - 0.618i)T^{2} \)
3 \( 1 + (-0.361 - 0.710i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (0.170 + 0.0869i)T + (4.11 + 5.66i)T^{2} \)
13 \( 1 + (-0.484 - 3.05i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (0.579 - 3.66i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (0.229 - 0.707i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.14 + 1.14i)T - 23iT^{2} \)
29 \( 1 + (-2.95 - 9.07i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.283 + 0.206i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.45 - 4.81i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (-6.36 - 2.06i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (3.72 + 3.72i)T + 43iT^{2} \)
47 \( 1 + (-11.0 + 5.61i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-8.91 + 1.41i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (9.15 - 2.97i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.46 + 4.76i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (4.13 + 4.13i)T + 67iT^{2} \)
71 \( 1 + (-9.27 + 6.73i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.09 - 2.14i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (0.542 + 0.394i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (16.4 + 2.60i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + 7.92iT - 89T^{2} \)
97 \( 1 + (-0.215 - 1.36i)T + (-92.2 + 29.9i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.04833735166185643674269401379, −10.67602774582213579282770128433, −10.00299736071807508547956080328, −9.119303700567846404378589950427, −8.602038519715714884045172304098, −7.23428288290431131271455093019, −6.55412680726004997464608344705, −4.65538361709771022487802314483, −3.75929523686197132965556842950, −1.54113562448905020168513791588, 0.950610613451803932372799106463, 2.59491478508666550953327890762, 4.41125406112176730878384123145, 5.77063240573519177981739253019, 7.22376845850755192343989152866, 7.966779485383532465855727055376, 8.835596519726818600799671753013, 9.713379635567130260979707557657, 10.66027490048885868418336287562, 11.39126384827554655967450477288

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