Properties

Label 2-3675-21.11-c0-0-14
Degree $2$
Conductor $3675$
Sign $0.980 + 0.198i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
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  + (1.22 − 0.707i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + 1.41·6-s + (0.499 + 0.866i)9-s + (0.866 − 0.5i)12-s + (0.499 + 0.866i)16-s + (1.22 + 0.707i)18-s + (−0.707 − 1.22i)19-s + (1.22 − 0.707i)23-s + 0.999i·27-s + (−0.707 + 1.22i)31-s + (1.22 + 0.707i)32-s + 0.999·36-s + (−1.73 − 0.999i)38-s + ⋯
L(s)  = 1  + (1.22 − 0.707i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + 1.41·6-s + (0.499 + 0.866i)9-s + (0.866 − 0.5i)12-s + (0.499 + 0.866i)16-s + (1.22 + 0.707i)18-s + (−0.707 − 1.22i)19-s + (1.22 − 0.707i)23-s + 0.999i·27-s + (−0.707 + 1.22i)31-s + (1.22 + 0.707i)32-s + 0.999·36-s + (−1.73 − 0.999i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.980 + 0.198i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.980 + 0.198i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.311516811\)
\(L(\frac12)\) \(\approx\) \(3.311516811\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.788128036740691156518365698954, −8.110522248286297790356248005331, −7.11166510290597170679073275995, −6.36140107302246899245015290607, −5.09662074588838118948360940883, −4.84499172053757079513917022053, −3.98511175287222552315677519644, −3.17908661713434603763101166161, −2.62136523608100298120299691394, −1.66811289812833002575451548184, 1.39731349845219405896615714256, 2.58717326025312167231728665623, 3.49238491922547449550816930259, 4.05697384397147086820536400699, 4.93869571461336062209580674995, 5.90568173503519435388187239101, 6.35187066612664268011829753564, 7.33024096150192318973061670521, 7.62401673251783917846562202827, 8.535188692756091546918445300540

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