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Properties

Label 2-3850-5.4-c1-0-18
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 − 3.23i·3-s − 4-s − 3.23·6-s − i·7-s + i·8-s − 7.47·9-s + 11-s + 3.23i·12-s + 1.23i·13-s − 14-s + 16-s + 6.47i·17-s + 7.47i·18-s + 2.76·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.86i·3-s − 0.5·4-s − 1.32·6-s − 0.377i·7-s + 0.353i·8-s − 2.49·9-s + 0.301·11-s + 0.934i·12-s + 0.342i·13-s − 0.267·14-s + 0.250·16-s + 1.56i·17-s + 1.76i·18-s + 0.634·19-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

\(L(1)\) \(\approx\) \(1.288420580\)
\(L(\frac12)\) \(\approx\) \(1.288420580\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.23iT - 3T^{2} \)
13 \( 1 - 1.23iT - 13T^{2} \)
17 \( 1 - 6.47iT - 17T^{2} \)
19 \( 1 - 2.76T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 10.9iT - 37T^{2} \)
41 \( 1 - 6.47T + 41T^{2} \)
43 \( 1 + 1.52iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + 0.472iT - 53T^{2} \)
59 \( 1 + 7.23T + 59T^{2} \)
61 \( 1 + 5.23T + 61T^{2} \)
67 \( 1 - 15.4iT - 67T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 + 4.94iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 3.52iT - 97T^{2} \)
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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

−8.227359210602016317324017755805, −7.71895336177814404590453646172, −6.89375078520049401602844358511, −6.26559574104444710459687645517, −5.59050799345235651764144532466, −4.43645495969863769291326520343, −3.40320638655436539252217643596, −2.59301432167620350666415976450, −1.53257578986277262814719196978, −1.11449608894753470508524368225, 0.42917313236932262962014904527, 2.64773738519335148180714783071, 3.30437871273578586488741997646, 4.32906079563726233520875573296, 4.79218568564416487079530247264, 5.52189541179605798737979062990, 6.12107857098069271303632033745, 7.16680783306737814128387567258, 8.044392298461557007322928285487, 8.810496139430503828366458435134

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