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Properties

Label 2-275-55.39-c2-0-24
Degree $2$
Conductor $275$
Sign $0.908 + 0.418i$
Analytic cond. $7.49320$
Root an. cond. $2.73737$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.224 + 0.690i)2-s + (0.812 − 1.11i)3-s + (2.80 − 2.04i)4-s + (0.954 + 0.310i)6-s + (8.05 − 5.85i)7-s + (4.39 + 3.19i)8-s + (2.19 + 6.74i)9-s + (−10.3 + 3.66i)11-s − 4.79i·12-s + (−1.62 − 5i)13-s + (5.85 + 4.25i)14-s + (3.07 − 9.45i)16-s + (4.72 − 14.5i)17-s + (−4.16 + 3.02i)18-s + (−1.21 + 1.67i)19-s + ⋯
L(s)  = 1  + (0.112 + 0.345i)2-s + (0.270 − 0.372i)3-s + (0.702 − 0.510i)4-s + (0.159 + 0.0517i)6-s + (1.15 − 0.836i)7-s + (0.549 + 0.398i)8-s + (0.243 + 0.749i)9-s + (−0.942 + 0.333i)11-s − 0.399i·12-s + (−0.124 − 0.384i)13-s + (0.418 + 0.303i)14-s + (0.192 − 0.591i)16-s + (0.277 − 0.854i)17-s + (−0.231 + 0.168i)18-s + (−0.0641 + 0.0882i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.908 + 0.418i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ 0.908 + 0.418i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.36432 - 0.517898i\)
\(L(\frac12)\) \(\approx\) \(2.36432 - 0.517898i\)
\(L(2)\) 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 \( 1 + (10.3 - 3.66i)T \)
good2 \( 1 + (-0.224 - 0.690i)T + (-3.23 + 2.35i)T^{2} \)
3 \( 1 + (-0.812 + 1.11i)T + (-2.78 - 8.55i)T^{2} \)
7 \( 1 + (-8.05 + 5.85i)T + (15.1 - 46.6i)T^{2} \)
13 \( 1 + (1.62 + 5i)T + (-136. + 99.3i)T^{2} \)
17 \( 1 + (-4.72 + 14.5i)T + (-233. - 169. i)T^{2} \)
19 \( 1 + (1.21 - 1.67i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + 2.76iT - 529T^{2} \)
29 \( 1 + (-16.7 - 22.9i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (2.20 + 6.77i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (23.6 + 32.5i)T + (-423. + 1.30e3i)T^{2} \)
41 \( 1 + (41.2 - 56.7i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 23.0T + 1.84e3T^{2} \)
47 \( 1 + (16.0 - 22.0i)T + (-682. - 2.10e3i)T^{2} \)
53 \( 1 + (-10.8 + 3.54i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (1.83 - 1.33i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-21.5 - 6.98i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 38.4iT - 4.48e3T^{2} \)
71 \( 1 + (-23.5 + 72.5i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (83.2 - 60.4i)T + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-3.74 + 1.21i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (25.7 - 79.1i)T + (-5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 123.T + 7.92e3T^{2} \)
97 \( 1 + (73.6 - 23.9i)T + (7.61e3 - 5.53e3i)T^{2} \)
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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

−11.39239114831089404786285806569, −10.68270817819083031840314178930, −10.02075785847046222121017003476, −8.253607134556709667708000917832, −7.58769523844623639445293157273, −6.97011241318270014982171710634, −5.37067160080848482644234370338, −4.70115091849040415003413541375, −2.61231262755275315351312884370, −1.37537857797470537739428359814, 1.81971185101800917130862943663, 3.04445218662854418353511537093, 4.28149853664341474929191869975, 5.57705226517666656507125669849, 6.85338503889387481771671761401, 8.085688987336694412069257576102, 8.661178727571576289892688248392, 10.04254915089453262118490830383, 10.87034563001730698908062802261, 11.84681834211719189172212072264

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