Properties

Label 2-3150-105.104-c1-0-7
Degree $2$
Conductor $3150$
Sign $-0.492 - 0.870i$
Analytic cond. $25.1528$
Root an. cond. $5.01526$
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  − 2-s + 4-s + (2.23 + 1.41i)7-s − 8-s + 1.41i·11-s − 0.926·13-s + (−2.23 − 1.41i)14-s + 16-s − 2.23i·17-s + 7.63i·19-s − 1.41i·22-s − 23-s + 0.926·26-s + (2.23 + 1.41i)28-s − 0.757i·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.845 + 0.534i)7-s − 0.353·8-s + 0.426i·11-s − 0.256·13-s + (−0.597 − 0.377i)14-s + 0.250·16-s − 0.542i·17-s + 1.75i·19-s − 0.301i·22-s − 0.208·23-s + 0.181·26-s + (0.422 + 0.267i)28-s − 0.140i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.492 - 0.870i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (3149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.492 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9483898453\)
\(L(\frac12)\) \(\approx\) \(0.9483898453\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.23 - 1.41i)T \)
good11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + 0.926T + 13T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
19 \( 1 - 7.63iT - 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + 0.757iT - 29T^{2} \)
31 \( 1 - 4.08iT - 31T^{2} \)
37 \( 1 + 2.82iT - 37T^{2} \)
41 \( 1 + 8.56T + 41T^{2} \)
43 \( 1 - 3.58iT - 43T^{2} \)
47 \( 1 - 1.30iT - 47T^{2} \)
53 \( 1 + 8.07T + 53T^{2} \)
59 \( 1 - 7.25T + 59T^{2} \)
61 \( 1 - 0.926iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 + 2.61T + 89T^{2} \)
97 \( 1 - 0.542T + 97T^{2} \)
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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

−8.819238795012160594496714381271, −8.211112509502681502036894091711, −7.62971377458995903848049873946, −6.85529702980237060723918229342, −5.90942847690246059860358145326, −5.22317643381144514232734228281, −4.31291440431150078416415210713, −3.19743576660060655518440465008, −2.12739128706677589010360492127, −1.37111067517537875049375871336, 0.37439406263420404213465104189, 1.53998293753685243949603595295, 2.53881126697437707771702740197, 3.61261201988250633661110299621, 4.62488176316346280855987206900, 5.33623213590807010676580988584, 6.41472676102494962913268374784, 7.04438619575927199140529529398, 7.81623099936799131156541678778, 8.418804261021071607415583862915

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