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Properties

Label 2-1075-43.42-c0-0-0
Degree $2$
Conductor $1075$
Sign $i$
Analytic cond. $0.536494$
Root an. cond. $0.732458$
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  + 1.80i·2-s + 1.24i·3-s − 2.24·4-s − 2.24·6-s + 0.445i·7-s − 2.24i·8-s − 0.554·9-s − 1.80·11-s − 2.80i·12-s − 0.801·14-s + 1.80·16-s − 0.999i·18-s − 0.554·21-s − 3.24i·22-s + 2.80·24-s + ⋯
L(s)  = 1  + 1.80i·2-s + 1.24i·3-s − 2.24·4-s − 2.24·6-s + 0.445i·7-s − 2.24i·8-s − 0.554·9-s − 1.80·11-s − 2.80i·12-s − 0.801·14-s + 1.80·16-s − 0.999i·18-s − 0.554·21-s − 3.24i·22-s + 2.80·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1075\)    =    \(5^{2} \cdot 43\)
Sign: $i$
Analytic conductor: \(0.536494\)
Root analytic conductor: \(0.732458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1075} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1075,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6634939241\)
\(L(\frac12)\) \(\approx\) \(0.6634939241\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
43 \( 1 - iT \)
good2 \( 1 - 1.80iT - T^{2} \)
3 \( 1 - 1.24iT - T^{2} \)
7 \( 1 - 0.445iT - T^{2} \)
11 \( 1 + 1.80T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 - 1.80iT - T^{2} \)
41 \( 1 - 1.24T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.24T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.24iT - T^{2} \)
79 \( 1 - 1.80T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
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

−10.32867146700373464402886844145, −9.647841524958325167346347644604, −8.928341753494076073158896120788, −8.100082328477754590950879626018, −7.51400412411326981069419812050, −6.38447203481127977390122596558, −5.46970363325612663038957594087, −4.99217376023858416339575941711, −4.19606706403958905713999108935, −2.90126437018762974186395805113, 0.60539309373857932924589515899, 1.97256196514299600070656801186, 2.62664650435063596584718568339, 3.78358661850419547895049813896, 4.88940508165112376347752991788, 5.87647691928723116924371639644, 7.31114906488850132420876215853, 7.81890171371019073049183814862, 8.806414814941717698862970515862, 9.718752954383947246132887906070

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