Properties

Label 2-1150-5.4-c3-0-75
Degree $2$
Conductor $1150$
Sign $-0.447 + 0.894i$
Analytic cond. $67.8521$
Root an. cond. $8.23724$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 7.57i·3-s − 4·4-s + 15.1·6-s − 35.4i·7-s + 8i·8-s − 30.3·9-s − 16.6·11-s − 30.2i·12-s + 79.9i·13-s − 70.8·14-s + 16·16-s − 46.8i·17-s + 60.6i·18-s + 110.·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.45i·3-s − 0.5·4-s + 1.03·6-s − 1.91i·7-s + 0.353i·8-s − 1.12·9-s − 0.455·11-s − 0.728i·12-s + 1.70i·13-s − 1.35·14-s + 0.250·16-s − 0.667i·17-s + 0.794i·18-s + 1.33·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(67.8521\)
Root analytic conductor: \(8.23724\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9260279340\)
\(L(\frac12)\) \(\approx\) \(0.9260279340\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
5 \( 1 \)
23 \( 1 - 23iT \)
good3 \( 1 - 7.57iT - 27T^{2} \)
7 \( 1 + 35.4iT - 343T^{2} \)
11 \( 1 + 16.6T + 1.33e3T^{2} \)
13 \( 1 - 79.9iT - 2.19e3T^{2} \)
17 \( 1 + 46.8iT - 4.91e3T^{2} \)
19 \( 1 - 110.T + 6.85e3T^{2} \)
29 \( 1 - 0.836T + 2.43e4T^{2} \)
31 \( 1 + 119.T + 2.97e4T^{2} \)
37 \( 1 - 368. iT - 5.06e4T^{2} \)
41 \( 1 + 95.7T + 6.89e4T^{2} \)
43 \( 1 + 331. iT - 7.95e4T^{2} \)
47 \( 1 + 535. iT - 1.03e5T^{2} \)
53 \( 1 + 409. iT - 1.48e5T^{2} \)
59 \( 1 - 352.T + 2.05e5T^{2} \)
61 \( 1 + 507.T + 2.26e5T^{2} \)
67 \( 1 + 820. iT - 3.00e5T^{2} \)
71 \( 1 + 733.T + 3.57e5T^{2} \)
73 \( 1 - 91.4iT - 3.89e5T^{2} \)
79 \( 1 + 329.T + 4.93e5T^{2} \)
83 \( 1 - 753. iT - 5.71e5T^{2} \)
89 \( 1 - 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + 271. iT - 9.12e5T^{2} \)
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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

−9.569139690985744304188041992709, −8.707094334547733989042182393891, −7.49129473953787681323008679730, −6.80472610124201597058869453172, −5.18961192725800259287628224432, −4.61562536715902462447582439170, −3.83502215060235030318799679925, −3.25604697081130000963720479443, −1.59776118079782908265763964897, −0.24437804750581416126602011241, 1.11306784095856652707275930997, 2.38004787532276485882699157876, 3.15971390247806440884629944594, 5.05146444221839506856218980300, 5.87739900138694887134677382959, 6.05348892924179661203965477898, 7.48703399183750746617326331212, 7.78707632598614402622744290096, 8.617281340469096438096066671429, 9.306571885096082012437473936958

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