Properties

Label 2-1950-65.64-c1-0-18
Degree $2$
Conductor $1950$
Sign $0.302 - 0.953i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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  + 2-s + i·3-s + 4-s + i·6-s + 4.56·7-s + 8-s − 9-s + 2.56i·11-s + i·12-s + (−3.56 + 0.561i)13-s + 4.56·14-s + 16-s + 5.68i·17-s − 18-s + 4.68i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 1.72·7-s + 0.353·8-s − 0.333·9-s + 0.772i·11-s + 0.288i·12-s + (−0.987 + 0.155i)13-s + 1.21·14-s + 0.250·16-s + 1.37i·17-s − 0.235·18-s + 1.07i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.302 - 0.953i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.302 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.134780398\)
\(L(\frac12)\) \(\approx\) \(3.134780398\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - iT \)
5 \( 1 \)
13 \( 1 + (3.56 - 0.561i)T \)
good7 \( 1 - 4.56T + 7T^{2} \)
11 \( 1 - 2.56iT - 11T^{2} \)
17 \( 1 - 5.68iT - 17T^{2} \)
19 \( 1 - 4.68iT - 19T^{2} \)
23 \( 1 + 5.12iT - 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + 1.43iT - 31T^{2} \)
37 \( 1 - 0.438T + 37T^{2} \)
41 \( 1 - 0.438iT - 41T^{2} \)
43 \( 1 - 2.87iT - 43T^{2} \)
47 \( 1 + 0.123T + 47T^{2} \)
53 \( 1 - 5iT - 53T^{2} \)
59 \( 1 + 5.43iT - 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 4.12T + 67T^{2} \)
71 \( 1 + 14.9iT - 71T^{2} \)
73 \( 1 + 1.12T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 - 3.12iT - 89T^{2} \)
97 \( 1 + 4.87T + 97T^{2} \)
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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

−9.448298775541371273926893435599, −8.247677853120741730432083161499, −7.976203878126542757912087228058, −6.93866683568875392276277256900, −5.92739318247823280687298326774, −5.06829470931469848649815236220, −4.50514807680803187153381606090, −3.86354345729159526014236897649, −2.41509112563544248808065595068, −1.63433480163886803347015552337, 0.924285400919668761143694720428, 2.15164465023259595211167077661, 2.94617678224004847199214692661, 4.25449104993498039730157823362, 5.24407233698133812719026404565, 5.38522889979832200756657608483, 6.83775924364765098792481916147, 7.34523999717992977688511822178, 8.076563003638453624339274561264, 8.825013793256585768837836394750

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