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Properties

Label 2-1205-5.4-c1-0-91
Degree $2$
Conductor $1205$
Sign $0.434 + 0.900i$
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  + 0.434i·2-s − 1.75i·3-s + 1.81·4-s + (0.971 + 2.01i)5-s + 0.763·6-s − 2.56i·7-s + 1.65i·8-s − 0.0918·9-s + (−0.874 + 0.422i)10-s − 2.96·11-s − 3.18i·12-s − 5.29i·13-s + 1.11·14-s + (3.54 − 1.70i)15-s + 2.90·16-s − 2.85i·17-s + ⋯
L(s)  = 1  + 0.307i·2-s − 1.01i·3-s + 0.905·4-s + (0.434 + 0.900i)5-s + 0.311·6-s − 0.967i·7-s + 0.585i·8-s − 0.0306·9-s + (−0.276 + 0.133i)10-s − 0.895·11-s − 0.919i·12-s − 1.46i·13-s + 0.297·14-s + (0.914 − 0.441i)15-s + 0.725·16-s − 0.693i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.434 + 0.900i$
Analytic conductor: \(9.62197\)
Root analytic conductor: \(3.10193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1205,\ (\ :1/2),\ 0.434 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.107352469\)
\(L(\frac12)\) \(\approx\) \(2.107352469\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.971 - 2.01i)T \)
241 \( 1 - T \)
good2 \( 1 - 0.434iT - 2T^{2} \)
3 \( 1 + 1.75iT - 3T^{2} \)
7 \( 1 + 2.56iT - 7T^{2} \)
11 \( 1 + 2.96T + 11T^{2} \)
13 \( 1 + 5.29iT - 13T^{2} \)
17 \( 1 + 2.85iT - 17T^{2} \)
19 \( 1 + 6.43T + 19T^{2} \)
23 \( 1 + 7.60iT - 23T^{2} \)
29 \( 1 - 7.01T + 29T^{2} \)
31 \( 1 - 3.60T + 31T^{2} \)
37 \( 1 + 0.899iT - 37T^{2} \)
41 \( 1 - 9.31T + 41T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 - 3.09iT - 47T^{2} \)
53 \( 1 + 3.14iT - 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 6.41T + 61T^{2} \)
67 \( 1 + 4.29iT - 67T^{2} \)
71 \( 1 + 9.51T + 71T^{2} \)
73 \( 1 - 7.02iT - 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 - 16.8iT - 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 - 8.44iT - 97T^{2} \)
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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

−9.964634904026863429113088725218, −8.179408224477311550064986275799, −7.86179115449184321533825396571, −6.98453919151955693889100144699, −6.53500604149855583790572764528, −5.80330351930080290222234895390, −4.48226738586826985374088398771, −2.81920680487115193530096930138, −2.44490393520334915934778141914, −0.886249164602748126763064873565, 1.69407746850552223129298990782, 2.48336934748033606447021015714, 3.87158813590354667175734409101, 4.66346382702128860769816877324, 5.63357494851191849400654106044, 6.32145327509696405843376419782, 7.45925994879901923901122923036, 8.663007242690544072316232786733, 9.052952968055967539949486181279, 10.10314238884812496552923248434

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