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Properties

Label 2-2175-87.86-c0-0-0
Degree $2$
Conductor $2175$
Sign $-0.707 - 0.707i$
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  − 1.41·2-s + (−0.707 + 0.707i)3-s + 1.00·4-s + (1.00 − 1.00i)6-s − 1.00i·9-s + (−0.707 + 0.707i)12-s − 0.999·16-s − 1.41·17-s + 1.41i·18-s + (0.707 + 0.707i)27-s + i·29-s + 1.41·32-s + 2.00·34-s − 1.00i·36-s + 1.41i·37-s + ⋯
L(s)  = 1  − 1.41·2-s + (−0.707 + 0.707i)3-s + 1.00·4-s + (1.00 − 1.00i)6-s − 1.00i·9-s + (−0.707 + 0.707i)12-s − 0.999·16-s − 1.41·17-s + 1.41i·18-s + (0.707 + 0.707i)27-s + i·29-s + 1.41·32-s + 2.00·34-s − 1.00i·36-s + 1.41i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2175\)    =    \(3 \cdot 5^{2} \cdot 29\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(1.08546\)
Root analytic conductor: \(1.04185\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2175} (1826, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2175,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2442097720\)
\(L(\frac12)\) \(\approx\) \(0.2442097720\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
29 \( 1 - iT \)
good2 \( 1 + 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 - 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 1.41iT - T^{2} \)
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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

−9.511385970112094172872074819385, −8.841176608305671191361704665744, −8.388602377731503583936217095417, −7.14327671169323356247640484887, −6.74406962298863786701846641547, −5.68376103870545192860691440655, −4.73682795611349426892386075174, −3.97060540279298605551049685533, −2.59021122765571353649079088826, −1.22802583446277363449727970081, 0.33228161875003712152339817706, 1.67862085979285519302565276236, 2.48864770213640067956031190971, 4.20865028647409363998692113396, 5.08215947008945072946602121854, 6.26222535343257359988756493995, 6.71032443837427956329442847953, 7.68767066791053322335997440709, 8.032248482562974332471705275280, 9.042372603365892433783286145829

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