Properties

Label 2-1075-43.42-c0-0-6
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  + 0.445i·2-s − 1.80i·3-s + 0.801·4-s + 0.801·6-s − 1.24i·7-s + 0.801i·8-s − 2.24·9-s − 0.445·11-s − 1.44i·12-s + 0.554·14-s + 0.445·16-s i·18-s − 2.24·21-s − 0.198i·22-s + 1.44·24-s + ⋯
L(s)  = 1  + 0.445i·2-s − 1.80i·3-s + 0.801·4-s + 0.801·6-s − 1.24i·7-s + 0.801i·8-s − 2.24·9-s − 0.445·11-s − 1.44i·12-s + 0.554·14-s + 0.445·16-s i·18-s − 2.24·21-s − 0.198i·22-s + 1.44·24-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

\(L(\frac{1}{2})\) \(\approx\) \(1.174059776\)
\(L(\frac12)\) \(\approx\) \(1.174059776\)
\(L(1)\) 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 \( 1 - 0.445iT - T^{2} \)
3 \( 1 + 1.80iT - T^{2} \)
7 \( 1 + 1.24iT - T^{2} \)
11 \( 1 + 0.445T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.24T + T^{2} \)
37 \( 1 - 0.445iT - T^{2} \)
41 \( 1 + 1.80T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.80T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.80iT - T^{2} \)
79 \( 1 - 0.445T + T^{2} \)
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

−10.04802328869678987004909067872, −8.485398056651046271666378763807, −7.931899453306413756839671331984, −7.28071855681999890402860631598, −6.70306150457009678744768019317, −6.09917469342986398595481584010, −4.96625506269522340318216423856, −3.28785505287123495977398712293, −2.23417876000048743471810051149, −1.13053787489434146946129715977, 2.30315987107524134514308809589, 3.06804972307304809422523897512, 3.99524390822339168955550715852, 5.15377866888966028538587975330, 5.71559322492703534722970612302, 6.76794693437643361978568397665, 8.198695376500249747867591820525, 8.834862171582189859377287698533, 9.765924483196383908707569251342, 10.22316143784553140164254018541

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