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Properties

Label 2-2175-435.434-c0-0-15
Degree $2$
Conductor $2175$
Sign $0.447 - 0.894i$
Analytic cond. $1.08546$
Root an. cond. $1.04185$
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  − 1.87i·2-s − i·3-s − 2.53·4-s − 1.87·6-s − 1.53i·7-s + 2.87i·8-s − 9-s − 0.347·11-s + 2.53i·12-s − 1.87i·13-s − 2.87·14-s + 2.87·16-s + 0.347i·17-s + 1.87i·18-s − 1.53·21-s + 0.652i·22-s + ⋯
L(s)  = 1  − 1.87i·2-s − i·3-s − 2.53·4-s − 1.87·6-s − 1.53i·7-s + 2.87i·8-s − 9-s − 0.347·11-s + 2.53i·12-s − 1.87i·13-s − 2.87·14-s + 2.87·16-s + 0.347i·17-s + 1.87i·18-s − 1.53·21-s + 0.652i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2175\)    =    \(3 \cdot 5^{2} \cdot 29\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(1.08546\)
Root analytic conductor: \(1.04185\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2175} (2174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2175,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7699113144\)
\(L(\frac12)\) \(\approx\) \(0.7699113144\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
29 \( 1 - T \)
good2 \( 1 + 1.87iT - T^{2} \)
7 \( 1 + 1.53iT - T^{2} \)
11 \( 1 + 0.347T + T^{2} \)
13 \( 1 + 1.87iT - T^{2} \)
17 \( 1 - 0.347iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.53iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 0.347iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 1.87T + T^{2} \)
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

−8.647937042684188351803014590194, −7.961737743587762001713005965490, −7.42376918599719405307017080864, −6.13378364356315335638082733306, −5.15863694830878826220903695214, −4.22692813553507140304874873045, −3.25624927978870964100807954002, −2.66794085219172610065449103803, −1.35548121006515235899879292803, −0.59307221157721873904594030647, 2.46164880092221341098034529901, 3.82006731852525174849727657097, 4.69381539196767019473620969287, 5.20672566668728024673586175719, 6.02033580658454346769357605457, 6.56914893171827255655266655194, 7.54605390217298435507639834430, 8.631133837735891416794324199487, 8.787358693339412918670150294845, 9.461594426662689311063538568132

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