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Properties

Label 2-756-21.20-c1-0-9
Degree $2$
Conductor $756$
Sign $0.654 + 0.755i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + 3·5-s + (2 − 1.73i)7-s − 5.19i·11-s − 3.46i·13-s − 6·17-s − 1.73i·19-s + 5.19i·23-s + 4·25-s + 10.3i·29-s − 5.19i·31-s + (6 − 5.19i)35-s + 37-s − 3·41-s + 10·43-s − 6·47-s + ⋯
L(s)  = 1  + 1.34·5-s + (0.755 − 0.654i)7-s − 1.56i·11-s − 0.960i·13-s − 1.45·17-s − 0.397i·19-s + 1.08i·23-s + 0.800·25-s + 1.92i·29-s − 0.933i·31-s + (1.01 − 0.878i)35-s + 0.164·37-s − 0.468·41-s + 1.52·43-s − 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79460 - 0.819863i\)
\(L(\frac12)\) \(\approx\) \(1.79460 - 0.819863i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 - 3T + 5T^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 - 5.19iT - 23T^{2} \)
29 \( 1 - 10.3iT - 29T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 - 5.19iT - 71T^{2} \)
73 \( 1 - 3.46iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−10.42496749245610712327378364349, −9.284623571176793489351541801442, −8.660084583323062691237669680778, −7.67606099440537164763174787528, −6.60749595961290575475892651466, −5.71135370211101716666465681475, −5.03837005126779033082726554224, −3.64674112267195386288722672565, −2.40930699420365285803489465925, −1.07219492169125979655317125175, 1.99298582318335137164776507068, 2.21162985299984596235187084605, 4.36652821479421606476640961845, 4.95729342157511820235378866601, 6.16478919125706584032999629009, 6.74501712182444563765675206637, 7.952537059707057697467903010867, 8.990673177544833188893339377022, 9.528051441513835824394691167351, 10.32568702344136244060203219218

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