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Properties

Label 2-3675-735.674-c0-0-0
Degree $2$
Conductor $3675$
Sign $-0.988 + 0.153i$
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  + (0.680 + 0.733i)3-s + (−0.365 + 0.930i)4-s + (−0.997 + 0.0747i)7-s + (−0.0747 + 0.997i)9-s + (−0.930 + 0.365i)12-s + (−0.829 + 1.72i)13-s + (−0.733 − 0.680i)16-s + (0.0747 − 0.129i)19-s + (−0.733 − 0.680i)21-s + (−0.781 + 0.623i)27-s + (0.294 − 0.955i)28-s + (−0.623 − 1.07i)31-s + (−0.900 − 0.433i)36-s + (1.84 − 0.722i)37-s + (−1.82 + 0.563i)39-s + ⋯
L(s)  = 1  + (0.680 + 0.733i)3-s + (−0.365 + 0.930i)4-s + (−0.997 + 0.0747i)7-s + (−0.0747 + 0.997i)9-s + (−0.930 + 0.365i)12-s + (−0.829 + 1.72i)13-s + (−0.733 − 0.680i)16-s + (0.0747 − 0.129i)19-s + (−0.733 − 0.680i)21-s + (−0.781 + 0.623i)27-s + (0.294 − 0.955i)28-s + (−0.623 − 1.07i)31-s + (−0.900 − 0.433i)36-s + (1.84 − 0.722i)37-s + (−1.82 + 0.563i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.988 + 0.153i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.988 + 0.153i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8111362859\)
\(L(\frac12)\) \(\approx\) \(0.8111362859\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.680 - 0.733i)T \)
5 \( 1 \)
7 \( 1 + (0.997 - 0.0747i)T \)
good2 \( 1 + (0.365 - 0.930i)T^{2} \)
11 \( 1 + (0.988 - 0.149i)T^{2} \)
13 \( 1 + (0.829 - 1.72i)T + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.955 + 0.294i)T^{2} \)
19 \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.955 - 0.294i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.84 + 0.722i)T + (0.733 - 0.680i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (1.75 + 0.400i)T + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.365 - 0.930i)T^{2} \)
53 \( 1 + (-0.733 - 0.680i)T^{2} \)
59 \( 1 + (-0.826 + 0.563i)T^{2} \)
61 \( 1 + (-0.266 - 0.680i)T + (-0.733 + 0.680i)T^{2} \)
67 \( 1 + (1.26 - 0.733i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (1.11 - 1.63i)T + (-0.365 - 0.930i)T^{2} \)
79 \( 1 + (-0.365 + 0.632i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.988 + 0.149i)T^{2} \)
97 \( 1 - 1.65iT - T^{2} \)
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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

−9.113836062131425663272666375471, −8.593223165375874253498612266774, −7.58337329862982734611415023978, −7.14954442744457586308686833389, −6.20181574030962073803339735373, −5.07291923141215410883507084305, −4.22896594503542940311428835593, −3.83354444133529401155849361170, −2.83215111997488231505388053776, −2.19226688621889827676823038325, 0.40992055149658821856707915502, 1.58919346438634634828461735723, 2.81671262156418147206064517528, 3.32691642876654943563647532458, 4.55406879813390610875167782307, 5.47343802829033773250957560674, 6.14541553012638402863614806099, 6.81439729377622293765469838839, 7.62215294038351913012733857132, 8.300072552547829739251811385436

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