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Properties

Label 2-3744-416.363-c0-0-3
Degree $2$
Conductor $3744$
Sign $-0.923 + 0.382i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
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  + (−0.980 − 0.195i)2-s + (0.923 + 0.382i)4-s + (0.360 + 0.149i)5-s + (−0.831 − 0.555i)8-s + (−0.324 − 0.216i)10-s + (−1.53 − 0.636i)11-s + (−0.382 − 0.923i)13-s + (0.707 + 0.707i)16-s + (0.275 + 0.275i)20-s + (1.38 + 0.923i)22-s + (−0.599 − 0.599i)25-s + (0.195 + 0.980i)26-s + (−0.555 − 0.831i)32-s + (−0.216 − 0.324i)40-s + (−1.38 − 1.38i)41-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)2-s + (0.923 + 0.382i)4-s + (0.360 + 0.149i)5-s + (−0.831 − 0.555i)8-s + (−0.324 − 0.216i)10-s + (−1.53 − 0.636i)11-s + (−0.382 − 0.923i)13-s + (0.707 + 0.707i)16-s + (0.275 + 0.275i)20-s + (1.38 + 0.923i)22-s + (−0.599 − 0.599i)25-s + (0.195 + 0.980i)26-s + (−0.555 − 0.831i)32-s + (−0.216 − 0.324i)40-s + (−1.38 − 1.38i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2701977879\)
\(L(\frac12)\) \(\approx\) \(0.2701977879\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.980 + 0.195i)T \)
3 \( 1 \)
13 \( 1 + (0.382 + 0.923i)T \)
good5 \( 1 + (-0.360 - 0.149i)T + (0.707 + 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (1.53 + 0.636i)T + (0.707 + 0.707i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
43 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
47 \( 1 - 1.11iT - T^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.750 - 1.81i)T + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.84T + T^{2} \)
83 \( 1 + (0.149 + 0.360i)T + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (-1.17 + 1.17i)T - iT^{2} \)
97 \( 1 - T^{2} \)
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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

−8.352861697327449182354129062320, −7.75979774988299473755706995272, −7.20783938915118165571706893537, −6.08448546150928062782397761440, −5.66882367971376029378188467916, −4.61763176124611609685578341258, −3.18314151269148288933984860177, −2.78145710963147660143214611249, −1.70820051307712964802692737023, −0.19295477679010845530618558627, 1.66313819059218628226822829482, 2.30624286798423139116859177947, 3.33883309491364075532214098548, 4.75232909435814852312654050111, 5.32057249130989943157395029919, 6.21152319660841082391038775010, 7.01804607711555918864019125042, 7.58179403386638355092711462210, 8.286583447068126107898976400270, 9.009155503871381272750515778281

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