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Properties

Label 2-275-55.24-c2-0-24
Degree $2$
Conductor $275$
Sign $-0.444 + 0.896i$
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 + (−2.35 − 3.23i)3-s + (2.80 + 2.04i)4-s + (−2.76 + 0.898i)6-s + (−0.224 − 0.163i)7-s + (4.39 − 3.19i)8-s + (−2.16 + 6.65i)9-s + (10.3 − 3.66i)11-s − 13.8i·12-s + (3.49 − 10.7i)13-s + (−0.163 + 0.118i)14-s + (3.07 + 9.45i)16-s + (−6.26 − 19.2i)17-s + (4.11 + 2.98i)18-s + (−16.9 − 23.3i)19-s + ⋯
L(s)  = 1  + (0.112 − 0.345i)2-s + (−0.783 − 1.07i)3-s + (0.702 + 0.510i)4-s + (−0.460 + 0.149i)6-s + (−0.0320 − 0.0233i)7-s + (0.549 − 0.398i)8-s + (−0.240 + 0.739i)9-s + (0.942 − 0.333i)11-s − 1.15i·12-s + (0.268 − 0.827i)13-s + (−0.0116 + 0.00846i)14-s + (0.192 + 0.591i)16-s + (−0.368 − 1.13i)17-s + (0.228 + 0.166i)18-s + (−0.893 − 1.22i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.444 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.444 + 0.896i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ -0.444 + 0.896i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.797433 - 1.28513i\)
\(L(\frac12)\) \(\approx\) \(0.797433 - 1.28513i\)
\(L(2)\) 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 \( 1 + (2.35 + 3.23i)T + (-2.78 + 8.55i)T^{2} \)
7 \( 1 + (0.224 + 0.163i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (-3.49 + 10.7i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (6.26 + 19.2i)T + (-233. + 169. i)T^{2} \)
19 \( 1 + (16.9 + 23.3i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 27.6iT - 529T^{2} \)
29 \( 1 + (10.2 - 14.0i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-10.6 + 32.6i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-7.80 + 10.7i)T + (-423. - 1.30e3i)T^{2} \)
41 \( 1 + (10.6 + 14.6i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 34.7T + 1.84e3T^{2} \)
47 \( 1 + (27.9 + 38.4i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (39.5 + 12.8i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-82.3 - 59.8i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-48.4 + 15.7i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 40.1iT - 4.48e3T^{2} \)
71 \( 1 + (-8.70 - 26.8i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-80.7 - 58.6i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-125. - 40.6i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-38.7 - 119. i)T + (-5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + 88.9T + 7.92e3T^{2} \)
97 \( 1 + (-45.9 - 14.9i)T + (7.61e3 + 5.53e3i)T^{2} \)
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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

−11.39680029685734235514568541370, −11.04433739958683137494784410917, −9.526980715733401508469431930005, −8.225665831254003779750314139033, −7.10330970995087406579357094047, −6.64465892522788601483533504598, −5.46777922278079238276985489068, −3.76943973718646503868571345568, −2.31737656316043028735024404336, −0.832077140266127751695800853089, 1.79722394690446140846753588602, 3.96826416058712152719713258263, 4.79616038509050711220048032473, 6.19417371860863151206372315255, 6.47264925850958938148968036971, 8.103991978441795251901841042243, 9.363783765078963385377254969491, 10.32781791901245978272429336283, 10.87750705555844905827001939458, 11.71988376913890893974598708741

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