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Properties

Label 2-2527-133.88-c0-0-0
Degree $2$
Conductor $2527$
Sign $-0.272 + 0.962i$
Analytic cond. $1.26113$
Root an. cond. $1.12300$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + 9-s + (0.5 − 0.866i)11-s + (−0.499 + 0.866i)16-s + 2·17-s − 0.999·20-s − 23-s + (−0.499 + 0.866i)28-s − 0.999·35-s + (−0.5 − 0.866i)36-s + (0.5 + 0.866i)43-s − 0.999·44-s + (0.5 − 0.866i)45-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + 9-s + (0.5 − 0.866i)11-s + (−0.499 + 0.866i)16-s + 2·17-s − 0.999·20-s − 23-s + (−0.499 + 0.866i)28-s − 0.999·35-s + (−0.5 − 0.866i)36-s + (0.5 + 0.866i)43-s − 0.999·44-s + (0.5 − 0.866i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $-0.272 + 0.962i$
Analytic conductor: \(1.26113\)
Root analytic conductor: \(1.12300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (1152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2527,\ (\ :0),\ -0.272 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.254162881\)
\(L(\frac12)\) \(\approx\) \(1.254162881\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - 2T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.094889738996092117335905584228, −8.157665599707246041679051490598, −7.38204634644112314853734095468, −6.31738536470293801415437336195, −5.80123847790584016256500026292, −4.94045820098014929819994212406, −4.15419779618485123924728425366, −3.35268911661649679635249293435, −1.51336363338379851387871481587, −0.978224034539823406766217833024, 1.76346385055665352594223022210, 2.82445315478393322423230371854, 3.57914796134275968253036637996, 4.43959860456829620859155288830, 5.47782799847112802279631780184, 6.31638849546079889634830163827, 7.12036365109043479879896475291, 7.68236527385182617986736640425, 8.552382183417946714269682844098, 9.572606047885828759493346665560

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