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Properties

Label 2-1815-5.4-c1-0-58
Degree $2$
Conductor $1815$
Sign $0.241 - 0.970i$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
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  + 1.53i·2-s − i·3-s − 0.369·4-s + (2.17 + 0.539i)5-s + 1.53·6-s + 0.290i·7-s + 2.51i·8-s − 9-s + (−0.829 + 3.34i)10-s + 0.369i·12-s − 6.97i·13-s − 0.447·14-s + (0.539 − 2.17i)15-s − 4.60·16-s + 4.78i·17-s − 1.53i·18-s + ⋯
L(s)  = 1  + 1.08i·2-s − 0.577i·3-s − 0.184·4-s + (0.970 + 0.241i)5-s + 0.628·6-s + 0.109i·7-s + 0.887i·8-s − 0.333·9-s + (−0.262 + 1.05i)10-s + 0.106i·12-s − 1.93i·13-s − 0.119·14-s + (0.139 − 0.560i)15-s − 1.15·16-s + 1.16i·17-s − 0.362i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.241 - 0.970i$
Analytic conductor: \(14.4928\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ 0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.373060245\)
\(L(\frac12)\) \(\approx\) \(2.373060245\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (-2.17 - 0.539i)T \)
11 \( 1 \)
good2 \( 1 - 1.53iT - 2T^{2} \)
7 \( 1 - 0.290iT - 7T^{2} \)
13 \( 1 + 6.97iT - 13T^{2} \)
17 \( 1 - 4.78iT - 17T^{2} \)
19 \( 1 - 7.75T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 7.41T + 29T^{2} \)
31 \( 1 - 6.34T + 31T^{2} \)
37 \( 1 - 3.41iT - 37T^{2} \)
41 \( 1 - 7.41T + 41T^{2} \)
43 \( 1 + 0.290iT - 43T^{2} \)
47 \( 1 + 5.26iT - 47T^{2} \)
53 \( 1 - 5.75iT - 53T^{2} \)
59 \( 1 + 3.60T + 59T^{2} \)
61 \( 1 - 6.68T + 61T^{2} \)
67 \( 1 + 6.15iT - 67T^{2} \)
71 \( 1 + 5.07T + 71T^{2} \)
73 \( 1 - 1.12iT - 73T^{2} \)
79 \( 1 + 0.921T + 79T^{2} \)
83 \( 1 + 1.70iT - 83T^{2} \)
89 \( 1 + 4.34T + 89T^{2} \)
97 \( 1 + 4.68iT - 97T^{2} \)
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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

−9.266628729254300117461823112976, −8.327895060186270712077670120051, −7.66800449360478882212970532168, −7.16537220243693997009522460304, −6.03734412555635897566382552551, −5.75341689450836023137391116210, −5.12486012461702985151094102768, −3.35917160604929187125475277808, −2.48181600999973805300676977255, −1.25587853126072217047620392194, 1.01781725162231601011668709410, 2.14226703114687434567043100894, 2.89545046993389894552480190875, 4.03316793998777281659075572231, 4.75875465007929563546123876387, 5.74539741918098101320367062390, 6.71216654779773861817881812962, 7.39381261156785909510866294742, 8.883350565312083502115241051745, 9.477325992216968582096089911276

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