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Properties

Label 2-2541-231.164-c0-0-3
Degree $2$
Conductor $2541$
Sign $0.774 + 0.632i$
Analytic cond. $1.26812$
Root an. cond. $1.12611$
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  + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.258 − 0.965i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s + 0.517·13-s + (−0.499 − 0.866i)16-s + (−0.707 − 1.22i)19-s + (0.258 − 0.965i)21-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (−0.965 − 0.258i)28-s + (−0.866 − 0.5i)31-s + 0.999·36-s + (0.866 + 1.5i)37-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.258 − 0.965i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s + 0.517·13-s + (−0.499 − 0.866i)16-s + (−0.707 − 1.22i)19-s + (0.258 − 0.965i)21-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (−0.965 − 0.258i)28-s + (−0.866 − 0.5i)31-s + 0.999·36-s + (0.866 + 1.5i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $0.774 + 0.632i$
Analytic conductor: \(1.26812\)
Root analytic conductor: \(1.12611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (1088, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :0),\ 0.774 + 0.632i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.787559429\)
\(L(\frac12)\) \(\approx\) \(1.787559429\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.258 + 0.965i)T \)
11 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - 0.517T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.93iT - T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−9.106857231520610577301139548795, −8.311964990467451010322108694046, −7.45392628828512920840159910525, −6.73634859485431629365376848559, −6.04907228844461449148962903239, −4.79872903325637314232144361571, −4.33240235549259138067695132560, −3.18875883984109171739125623878, −2.37158287278356327046484828911, −1.14170963451098918815630934513, 1.77663919205478782170928055112, 2.44895252372201806151136604707, 3.47068260540940188681416785754, 3.92037935787161410646234953778, 5.45012623381217646680622977957, 6.27407719477535028032281209746, 7.00604132528163165442120674453, 7.70908850625855022064555638634, 8.494435954145039616909919749879, 8.846938469574129937829679245172

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