Properties

Label 2-3850-5.4-c1-0-85
Degree $2$
Conductor $3850$
Sign $-0.447 - 0.894i$
Analytic cond. $30.7424$
Root an. cond. $5.54458$
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  i·2-s − 2i·3-s − 4-s − 2·6-s i·7-s + i·8-s − 9-s − 11-s + 2i·12-s + 2i·13-s − 14-s + 16-s − 6i·17-s + i·18-s − 2·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.15i·3-s − 0.5·4-s − 0.816·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 0.301·11-s + 0.577i·12-s + 0.554i·13-s − 0.267·14-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s − 0.458·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3850\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(30.7424\)
Root analytic conductor: \(5.54458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3850} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3850,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.056582103\)
\(L(\frac12)\) \(\approx\) \(1.056582103\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
7 \( 1 + iT \)
11 \( 1 + T \)
good3 \( 1 + 2iT - 3T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−7.897894457760361382740339582380, −7.29935467567606239808091724305, −6.64528148177060704380301008250, −5.87955401896915691480635373505, −4.71980815172921511580929961786, −4.21755996169935317041061406748, −2.87810699699757592728356600876, −2.31580726865117764745013998178, −1.25194369111952391920379542501, −0.32414433511706575374543522213, 1.54959607914988530110163358505, 3.03290077045981257801998467595, 3.73134471099496489293692804294, 4.62626840556589829670061105187, 5.12961404772351430051208950385, 6.03739051632072380422913906033, 6.53698197148980954081587392981, 7.81840832332704904083289543596, 8.110100837725497224495687874140, 9.042254935673947115081976576770

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