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Properties

Label 2-1815-5.4-c1-0-105
Degree $2$
Conductor $1815$
Sign $0.339 - 0.940i$
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  − 1.90i·2-s − i·3-s − 1.63·4-s + (−2.10 − 0.759i)5-s − 1.90·6-s − 4.45i·7-s − 0.689i·8-s − 9-s + (−1.44 + 4.01i)10-s + 1.63i·12-s − 0.534i·13-s − 8.50·14-s + (−0.759 + 2.10i)15-s − 4.59·16-s − 4.54i·17-s + 1.90i·18-s + ⋯
L(s)  = 1  − 1.34i·2-s − 0.577i·3-s − 0.819·4-s + (−0.940 − 0.339i)5-s − 0.778·6-s − 1.68i·7-s − 0.243i·8-s − 0.333·9-s + (−0.458 + 1.26i)10-s + 0.472i·12-s − 0.148i·13-s − 2.27·14-s + (−0.196 + 0.543i)15-s − 1.14·16-s − 1.10i·17-s + 0.449i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.339 - 0.940i$
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: \((2,\ 1815,\ (\ :1/2),\ 0.339 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.088317674\)
\(L(\frac12)\) \(\approx\) \(1.088317674\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 + 0.759i)T \)
11 \( 1 \)
good2 \( 1 + 1.90iT - 2T^{2} \)
7 \( 1 + 4.45iT - 7T^{2} \)
13 \( 1 + 0.534iT - 13T^{2} \)
17 \( 1 + 4.54iT - 17T^{2} \)
19 \( 1 + 1.92T + 19T^{2} \)
23 \( 1 + 5.18iT - 23T^{2} \)
29 \( 1 - 4.75T + 29T^{2} \)
31 \( 1 - 3.49T + 31T^{2} \)
37 \( 1 - 0.527iT - 37T^{2} \)
41 \( 1 - 4.69T + 41T^{2} \)
43 \( 1 - 3.02iT - 43T^{2} \)
47 \( 1 + 11.9iT - 47T^{2} \)
53 \( 1 - 5.02iT - 53T^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 11.5iT - 67T^{2} \)
71 \( 1 - 3.36T + 71T^{2} \)
73 \( 1 + 0.650iT - 73T^{2} \)
79 \( 1 - 17.7T + 79T^{2} \)
83 \( 1 - 4.24iT - 83T^{2} \)
89 \( 1 - 8.24T + 89T^{2} \)
97 \( 1 + 1.13iT - 97T^{2} \)
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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

−8.664547937913167427506593148915, −7.87284061799762143829010878078, −7.09338963630810054498073917959, −6.56671196655624574966553891885, −4.83987159414787319235329695055, −4.24698805254726842611343401977, −3.43376446085511890033590548898, −2.49893229466947597139912893908, −1.04794636483332289330658292356, −0.48364056397775644439062612742, 2.24567274364261387481311061479, 3.30955189332588220881933250341, 4.43566214132973136069426000652, 5.19182215948237087988078337548, 6.08379122115542248145626835708, 6.49791477963117316771177494055, 7.71425559549714222125188425644, 8.198129592170724594620585997477, 8.847464585881020035891716763713, 9.500916400452282062531828711604

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