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Properties

Label 2-855-285.113-c0-0-3
Degree $2$
Conductor $855$
Sign $0.927 + 0.374i$
Analytic cond. $0.426700$
Root an. cond. $0.653223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·4-s + (−0.707 − 0.707i)5-s + (1 − i)7-s − 1.41i·11-s − 16-s + (1.41 + 1.41i)17-s − i·19-s + (0.707 − 0.707i)20-s + 1.00i·25-s + (1 + i)28-s − 1.41·35-s + (1 + i)43-s + 1.41·44-s − i·49-s + (−1.00 + 1.00i)55-s + ⋯
L(s)  = 1  + i·4-s + (−0.707 − 0.707i)5-s + (1 − i)7-s − 1.41i·11-s − 16-s + (1.41 + 1.41i)17-s − i·19-s + (0.707 − 0.707i)20-s + 1.00i·25-s + (1 + i)28-s − 1.41·35-s + (1 + i)43-s + 1.41·44-s − i·49-s + (−1.00 + 1.00i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ 0.927 + 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9771947645\)
\(L(\frac12)\) \(\approx\) \(0.9771947645\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + iT \)
good2 \( 1 - iT^{2} \)
7 \( 1 + (-1 + i)T - iT^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.66265702829090194013764499902, −9.215022136866356169879384352250, −8.339049976797071981307955963554, −7.965716139537792892574275884698, −7.27759840875288794519792733503, −5.93026358900725877888260301653, −4.73316779243943655341063854976, −3.96890937117038100207801372305, −3.16587294602549030627876714434, −1.18520205959362365058518248948, 1.68239202999214222943519702195, 2.78386326967141736761795840307, 4.32743402905955723702091713024, 5.17609792733199447199423785188, 5.93750130621744318739237326962, 7.19024654118284638211386858903, 7.69332947336015004269514940348, 8.821753467486876688449127119671, 9.774036768593061816985028948567, 10.33506105692987374052171582985

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