Properties

Label 2-245-1.1-c3-0-6
Degree $2$
Conductor $245$
Sign $1$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 2·3-s + 8·4-s + 5·5-s + 8·6-s − 23·9-s − 20·10-s + 32·11-s − 16·12-s + 38·13-s − 10·15-s − 64·16-s − 26·17-s + 92·18-s − 100·19-s + 40·20-s − 128·22-s − 78·23-s + 25·25-s − 152·26-s + 100·27-s − 50·29-s + 40·30-s + 108·31-s + 256·32-s − 64·33-s + 104·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.384·3-s + 4-s + 0.447·5-s + 0.544·6-s − 0.851·9-s − 0.632·10-s + 0.877·11-s − 0.384·12-s + 0.810·13-s − 0.172·15-s − 16-s − 0.370·17-s + 1.20·18-s − 1.20·19-s + 0.447·20-s − 1.24·22-s − 0.707·23-s + 1/5·25-s − 1.14·26-s + 0.712·27-s − 0.320·29-s + 0.243·30-s + 0.625·31-s + 1.41·32-s − 0.337·33-s + 0.524·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{245} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7117047454\)
\(L(\frac12)\) \(\approx\) \(0.7117047454\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - p T \)
7 \( 1 \)
good2 \( 1 + p^{2} T + p^{3} T^{2} \)
3 \( 1 + 2 T + p^{3} T^{2} \)
11 \( 1 - 32 T + p^{3} T^{2} \)
13 \( 1 - 38 T + p^{3} T^{2} \)
17 \( 1 + 26 T + p^{3} T^{2} \)
19 \( 1 + 100 T + p^{3} T^{2} \)
23 \( 1 + 78 T + p^{3} T^{2} \)
29 \( 1 + 50 T + p^{3} T^{2} \)
31 \( 1 - 108 T + p^{3} T^{2} \)
37 \( 1 - 266 T + p^{3} T^{2} \)
41 \( 1 + 22 T + p^{3} T^{2} \)
43 \( 1 - 442 T + p^{3} T^{2} \)
47 \( 1 - 514 T + p^{3} T^{2} \)
53 \( 1 - 2 T + p^{3} T^{2} \)
59 \( 1 + 500 T + p^{3} T^{2} \)
61 \( 1 - 518 T + p^{3} T^{2} \)
67 \( 1 - 126 T + p^{3} T^{2} \)
71 \( 1 - 412 T + p^{3} T^{2} \)
73 \( 1 - 878 T + p^{3} T^{2} \)
79 \( 1 - 600 T + p^{3} T^{2} \)
83 \( 1 + 282 T + p^{3} T^{2} \)
89 \( 1 - 150 T + p^{3} T^{2} \)
97 \( 1 + 386 T + p^{3} T^{2} \)
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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

−11.17770147747942185827783866724, −10.71326176350318202669320425329, −9.539615985882177534218290465943, −8.829969089638996354733200989139, −8.052634846466306654893111669322, −6.66099074413071627698497576010, −5.91281441915330040453203797521, −4.21700883997275093707582999933, −2.24428123059458504388370733691, −0.77784690605546061366834461245, 0.77784690605546061366834461245, 2.24428123059458504388370733691, 4.21700883997275093707582999933, 5.91281441915330040453203797521, 6.66099074413071627698497576010, 8.052634846466306654893111669322, 8.829969089638996354733200989139, 9.539615985882177534218290465943, 10.71326176350318202669320425329, 11.17770147747942185827783866724

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