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Properties

Label 2-245-5.4-c1-0-9
Degree $2$
Conductor $245$
Sign $0.447 + 0.894i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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 − i·3-s + 4-s + (1 + 2i)5-s − 6-s − 3i·8-s + 2·9-s + (2 − i)10-s − i·12-s + 2i·13-s + (2 − i)15-s − 16-s − 2i·17-s − 2i·18-s − 6·19-s + (1 + 2i)20-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s + 0.5·4-s + (0.447 + 0.894i)5-s − 0.408·6-s − 1.06i·8-s + 0.666·9-s + (0.632 − 0.316i)10-s − 0.288i·12-s + 0.554i·13-s + (0.516 − 0.258i)15-s − 0.250·16-s − 0.485i·17-s − 0.471i·18-s − 1.37·19-s + (0.223 + 0.447i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32957 - 0.821725i\)
\(L(\frac12)\) \(\approx\) \(1.32957 - 0.821725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1 - 2i)T \)
7 \( 1 \)
good2 \( 1 + iT - 2T^{2} \)
3 \( 1 + iT - 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 7iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 - 5iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 11iT - 83T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−11.84681627409977531747499048886, −10.98756286558056088332578216807, −10.25327295647993626477769852673, −9.351854607884083566221417858624, −7.73218485914141313641570698096, −6.80122903032974038134729394574, −6.21699066979898812751951526873, −4.22970151780495777256668162325, −2.75196144789082061150707603900, −1.71885448512894583614551319232, 1.96757377625627144750002423327, 3.99128029617419054937583513285, 5.22034242659032869267184379667, 6.06457335607048116366724981930, 7.29763084938530033112741735749, 8.315789301112807317702033567314, 9.271730083406293209770252837771, 10.32375677978690018736264978442, 11.13252314025887071708034118484, 12.46951940122934231732586921988

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