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Properties

Label 1-1925-1925.1173-r0-0-0
Degree $1$
Conductor $1925$
Sign $-0.400 + 0.916i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
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  + (−0.866 − 0.5i)2-s + (0.743 + 0.669i)3-s + (0.5 + 0.866i)4-s + (−0.309 − 0.951i)6-s − i·8-s + (0.104 + 0.994i)9-s + (−0.207 + 0.978i)12-s + (0.587 + 0.809i)13-s + (−0.5 + 0.866i)16-s + (0.743 + 0.669i)17-s + (0.406 − 0.913i)18-s + (−0.5 + 0.866i)19-s + (−0.743 + 0.669i)23-s + (0.669 − 0.743i)24-s + (−0.104 − 0.994i)26-s + (−0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.743 + 0.669i)3-s + (0.5 + 0.866i)4-s + (−0.309 − 0.951i)6-s − i·8-s + (0.104 + 0.994i)9-s + (−0.207 + 0.978i)12-s + (0.587 + 0.809i)13-s + (−0.5 + 0.866i)16-s + (0.743 + 0.669i)17-s + (0.406 − 0.913i)18-s + (−0.5 + 0.866i)19-s + (−0.743 + 0.669i)23-s + (0.669 − 0.743i)24-s + (−0.104 − 0.994i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.400 + 0.916i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ -0.400 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6517780224 + 0.9957346314i\)
\(L(\frac12)\) \(\approx\) \(0.6517780224 + 0.9957346314i\)
\(L(1)\) \(\approx\) \(0.8717392442 + 0.2668836338i\)
\(L(1)\) \(\approx\) \(0.8717392442 + 0.2668836338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.743 + 0.669i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (0.743 + 0.669i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.743 + 0.669i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.406 - 0.913i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (-0.207 - 0.978i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (0.406 - 0.913i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.743 - 0.669i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (0.951 - 0.309i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.604444255712353985914989844010, −19.02799009779317709687903578749, −18.26514905206951687743670736001, −17.86718101526889381681400518778, −17.01106885587312323923442473528, −16.14572134497125780208570251437, −15.36245855581627902569621551537, −14.85784224725335529349269935472, −13.92795388070258189269235837133, −13.43270122724805836257145195443, −12.35827613245238052084662406204, −11.65309625805697479724409181469, −10.6170933569671564146441732441, −9.93158782287858366184179388558, −9.092894115163851447562347393340, −8.383643930305705761514191383159, −7.87096577821383942701954341161, −7.04102434751204341721221141228, −6.3443690376088962649637652713, −5.56647106731793315966057166930, −4.406970705415279074443297585955, −3.10154250501852223733239871878, −2.442892679705652342962963078888, −1.37615847616946358304341829077, −0.502667558947458914070557043129, 1.45593295893810225094855400310, 2.01324335578501152448089769034, 3.21314386607339159862638197704, 3.721657687216904397532444894369, 4.55567894384462749597959213302, 5.847173049737924126937109002, 6.80698528354610573979761966682, 7.84352177005552927228913575509, 8.343643950869324815697459983004, 9.045733314870520378141815549547, 9.83276949371884705555748942738, 10.40342327225999059734178231838, 11.08724606840462551507856745315, 12.031626731858912363938735971370, 12.72164413082110637924795468572, 13.75964975138793951677620753302, 14.354384798654566840248683851563, 15.30199775742114306465529999447, 16.13485429456368789925382146530, 16.46539716585479926890425862723, 17.39294200241280796531358672796, 18.20060648288757298751495949027, 19.14244573763128050459063978487, 19.36206776511536627924442187569, 20.24340459635692359295232738263

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