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Properties

Label 2-1305-5.4-c1-0-21
Degree $2$
Conductor $1305$
Sign $0.894 + 0.447i$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
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  − 2i·2-s − 2·4-s + (2 + i)5-s + 2i·7-s + (2 − 4i)10-s + 3·11-s + 4i·13-s + 4·14-s − 4·16-s + 8i·17-s + (−4 − 2i)20-s − 6i·22-s + i·23-s + (3 + 4i)25-s + 8·26-s + ⋯
L(s)  = 1  − 1.41i·2-s − 4-s + (0.894 + 0.447i)5-s + 0.755i·7-s + (0.632 − 1.26i)10-s + 0.904·11-s + 1.10i·13-s + 1.06·14-s − 16-s + 1.94i·17-s + (−0.894 − 0.447i)20-s − 1.27i·22-s + 0.208i·23-s + (0.600 + 0.800i)25-s + 1.56·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.895737640\)
\(L(\frac12)\) \(\approx\) \(1.895737640\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2 - i)T \)
29 \( 1 - T \)
good2 \( 1 + 2iT - 2T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 8iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - 9iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 13iT - 97T^{2} \)
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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

−9.757830890324680421257500645776, −9.087984649008141095164856247320, −8.431790346998482419687020396985, −6.80125036026447721708620469006, −6.36868223560415642204125320538, −5.27356761140796604447676436149, −4.02770147782157486020210128698, −3.31025327241847479150162293105, −1.98979331973206540154965494133, −1.66837869397666754533906565074, 0.808983604750814900464099448857, 2.46081331880117024809509595742, 3.93877574367662835515018825413, 5.09538745232020226661396787702, 5.50795266801216663163731395857, 6.52940572166120069586844787306, 7.16829402594956136890185706198, 7.83505957045947584722038934808, 8.943330686626200754391810227476, 9.304651197910903090142827882416

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