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Properties

Label 2-275-11.3-c1-0-6
Degree $2$
Conductor $275$
Sign $0.698 - 0.716i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.06 − 0.776i)2-s + (0.626 + 1.92i)3-s + (−0.0782 + 0.240i)4-s + (2.16 + 1.57i)6-s + (−0.107 + 0.331i)7-s + (0.920 + 2.83i)8-s + (−0.895 + 0.650i)9-s + (−3.25 + 0.638i)11-s − 0.513·12-s + (1.50 − 1.09i)13-s + (0.142 + 0.437i)14-s + (2.77 + 2.01i)16-s + (1.45 + 1.05i)17-s + (−0.451 + 1.39i)18-s + (−2.21 − 6.80i)19-s + ⋯
L(s)  = 1  + (0.756 − 0.549i)2-s + (0.361 + 1.11i)3-s + (−0.0391 + 0.120i)4-s + (0.884 + 0.642i)6-s + (−0.0406 + 0.125i)7-s + (0.325 + 1.00i)8-s + (−0.298 + 0.216i)9-s + (−0.981 + 0.192i)11-s − 0.148·12-s + (0.416 − 0.302i)13-s + (0.0380 + 0.116i)14-s + (0.693 + 0.503i)16-s + (0.353 + 0.256i)17-s + (−0.106 + 0.327i)18-s + (−0.507 − 1.56i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.698 - 0.716i$
Analytic conductor: \(2.19588\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1/2),\ 0.698 - 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81612 + 0.765908i\)
\(L(\frac12)\) \(\approx\) \(1.81612 + 0.765908i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.25 - 0.638i)T \)
good2 \( 1 + (-1.06 + 0.776i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.626 - 1.92i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.107 - 0.331i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.50 + 1.09i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.45 - 1.05i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.21 + 6.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 8.41T + 23T^{2} \)
29 \( 1 + (-1.79 + 5.51i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.49 - 3.26i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.571 + 1.75i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.79 + 8.59i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.76T + 43T^{2} \)
47 \( 1 + (2.24 + 6.90i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.277 - 0.201i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.0615 + 0.189i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.30 - 4.58i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + (7.95 + 5.77i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.313 + 0.963i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.36 - 3.89i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.00 - 6.53i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.84T + 89T^{2} \)
97 \( 1 + (5.01 - 3.64i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.06532092706802227199588583617, −10.90135765091744063530938194170, −10.47342150169639760545555037481, −9.155545967317637545403676008425, −8.456240691643771839572267928744, −7.14219431969068267925291109871, −5.35720413021240572964373035158, −4.65495174269892907110455029012, −3.54067705731635243950754054005, −2.61967156554123770402127234451, 1.43154947065627424516731241113, 3.22384135060527533206259336101, 4.75014345410641211085695288582, 5.83423163527688137911830469880, 6.80254466202123137000937765691, 7.59269090284750277051259083944, 8.562916034554626619492334302786, 9.933649202401003062657761670278, 10.85550905001197199676839976894, 12.22669894199014387562266209132

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