Properties

Label 2-1850-5.4-c1-0-49
Degree $2$
Conductor $1850$
Sign $-0.894 - 0.447i$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
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  i·2-s − 1.53i·3-s − 4-s − 1.53·6-s − 2.87i·7-s + i·8-s + 0.630·9-s − 1.09·11-s + 1.53i·12-s − 4.53i·13-s − 2.87·14-s + 16-s + 2.80i·17-s − 0.630i·18-s + 5.04·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.888i·3-s − 0.5·4-s − 0.628·6-s − 1.08i·7-s + 0.353i·8-s + 0.210·9-s − 0.329·11-s + 0.444i·12-s − 1.25i·13-s − 0.769·14-s + 0.250·16-s + 0.679i·17-s − 0.148i·18-s + 1.15·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(14.7723\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1850} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.367934458\)
\(L(\frac12)\) \(\approx\) \(1.367934458\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
37 \( 1 - iT \)
good3 \( 1 + 1.53iT - 3T^{2} \)
7 \( 1 + 2.87iT - 7T^{2} \)
11 \( 1 + 1.09T + 11T^{2} \)
13 \( 1 + 4.53iT - 13T^{2} \)
17 \( 1 - 2.80iT - 17T^{2} \)
19 \( 1 - 5.04T + 19T^{2} \)
23 \( 1 + 7.41iT - 23T^{2} \)
29 \( 1 + 6.68T + 29T^{2} \)
31 \( 1 - 3.51T + 31T^{2} \)
41 \( 1 + 8.07T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 - 8.68iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 - 6.29T + 61T^{2} \)
67 \( 1 - 13.2iT - 67T^{2} \)
71 \( 1 - 6.29T + 71T^{2} \)
73 \( 1 + 12.7iT - 73T^{2} \)
79 \( 1 + 2.58T + 79T^{2} \)
83 \( 1 + 8.48iT - 83T^{2} \)
89 \( 1 - 6.51T + 89T^{2} \)
97 \( 1 - 3.07iT - 97T^{2} \)
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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

−8.719205433711584217520713483751, −7.83755960583173033190996020582, −7.45037522205257187897049266835, −6.50522988321147827929697847086, −5.53358752039962179780261788113, −4.51829684238159784141421063304, −3.61322112380738309964549347529, −2.63219155795501243909286484836, −1.42400108304827370403077548257, −0.53370545389602944928403882650, 1.74387961788892962886322274658, 3.16547391109303305961614879876, 4.02925204425138839554718935885, 5.16087911403662065132280884274, 5.33764509727001386294278482465, 6.54603635400240856125697308244, 7.30059070171358478582268189049, 8.141975143802460841840960808908, 9.189117603180988001893832540409, 9.439478341381714955986585437475

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