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Properties

Label 2-1205-241.240-c1-0-8
Degree $2$
Conductor $1205$
Sign $-0.975 + 0.222i$
Analytic cond. $9.62197$
Root an. cond. $3.10193$
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  − 0.563·2-s − 1.90·3-s − 1.68·4-s − 5-s + 1.07·6-s + 5.09i·7-s + 2.07·8-s + 0.626·9-s + 0.563·10-s − 1.45i·11-s + 3.20·12-s + 5.62i·13-s − 2.87i·14-s + 1.90·15-s + 2.19·16-s + 3.13i·17-s + ⋯
L(s)  = 1  − 0.398·2-s − 1.09·3-s − 0.841·4-s − 0.447·5-s + 0.438·6-s + 1.92i·7-s + 0.733·8-s + 0.208·9-s + 0.178·10-s − 0.438i·11-s + 0.925·12-s + 1.55i·13-s − 0.767i·14-s + 0.491·15-s + 0.549·16-s + 0.760i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.975 + 0.222i$
Analytic conductor: \(9.62197\)
Root analytic conductor: \(3.10193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1205,\ (\ :1/2),\ -0.975 + 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3816354041\)
\(L(\frac12)\) \(\approx\) \(0.3816354041\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
241 \( 1 + (-15.1 + 3.44i)T \)
good2 \( 1 + 0.563T + 2T^{2} \)
3 \( 1 + 1.90T + 3T^{2} \)
7 \( 1 - 5.09iT - 7T^{2} \)
11 \( 1 + 1.45iT - 11T^{2} \)
13 \( 1 - 5.62iT - 13T^{2} \)
17 \( 1 - 3.13iT - 17T^{2} \)
19 \( 1 - 2.89iT - 19T^{2} \)
23 \( 1 + 2.22iT - 23T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 - 7.65iT - 31T^{2} \)
37 \( 1 - 5.92iT - 37T^{2} \)
41 \( 1 - 4.55T + 41T^{2} \)
43 \( 1 - 5.80iT - 43T^{2} \)
47 \( 1 - 0.164T + 47T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 + 8.43T + 59T^{2} \)
61 \( 1 + 2.65T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 - 9.18iT - 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 + 7.61T + 83T^{2} \)
89 \( 1 + 0.929iT - 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−10.13854711794805652574197511612, −9.203272958668990639266219866083, −8.607370036275428893932165877851, −8.108279044394669961134973296370, −6.52641469272407923160400456429, −6.07115272335946691567840688623, −5.06091477260194629401676764604, −4.48887160681267676512176627565, −3.05917272033650812334627799329, −1.50871773485216709656481800530, 0.34191513497371791679070575943, 0.855412428239293838949891997987, 3.23660066896181608098934741125, 4.35365506901757408173526050859, 4.84882577617675006985758071464, 5.85340120741265569743534647510, 7.06757004655942203725717540897, 7.58886786318228694148683565111, 8.333162812681349005641848329788, 9.586827993982163741311749909056

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