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Properties

Label 2-1850-5.4-c1-0-28
Degree $2$
Conductor $1850$
Sign $0.447 - 0.894i$
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 − 0.406i·3-s − 4-s + 0.406·6-s + 2.91i·7-s − i·8-s + 2.83·9-s + 6.51·11-s + 0.406i·12-s + 0.813i·13-s − 2.91·14-s + 16-s − 2.51i·17-s + 2.83i·18-s − 0.406·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.234i·3-s − 0.5·4-s + 0.166·6-s + 1.10i·7-s − 0.353i·8-s + 0.944·9-s + 1.96·11-s + 0.117i·12-s + 0.225i·13-s − 0.779·14-s + 0.250·16-s − 0.608i·17-s + 0.668i·18-s − 0.0933·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $0.447 - 0.894i$
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: \((2,\ 1850,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.024357933\)
\(L(\frac12)\) \(\approx\) \(2.024357933\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.406iT - 3T^{2} \)
7 \( 1 - 2.91iT - 7T^{2} \)
11 \( 1 - 6.51T + 11T^{2} \)
13 \( 1 - 0.813iT - 13T^{2} \)
17 \( 1 + 2.51iT - 17T^{2} \)
19 \( 1 + 0.406T + 19T^{2} \)
23 \( 1 + 5.02iT - 23T^{2} \)
29 \( 1 - 5.32T + 29T^{2} \)
31 \( 1 + 8.75T + 31T^{2} \)
41 \( 1 - 6.34T + 41T^{2} \)
43 \( 1 - 7.32iT - 43T^{2} \)
47 \( 1 + 5.42iT - 47T^{2} \)
53 \( 1 - 2.34iT - 53T^{2} \)
59 \( 1 - 1.42T + 59T^{2} \)
61 \( 1 + 1.32T + 61T^{2} \)
67 \( 1 + 5.42iT - 67T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 - 11.0iT - 73T^{2} \)
79 \( 1 - 1.75T + 79T^{2} \)
83 \( 1 - 7.05iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 2.34iT - 97T^{2} \)
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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

−9.228470205736900585774089090771, −8.693889286351439285529710443431, −7.74385933721146117816726907187, −6.72473783347739364742459170913, −6.51522630809358657190874370344, −5.47659763544081397398694810382, −4.50215421854489219243444353815, −3.77151722058445054946407084986, −2.34386161067995139994168502255, −1.13456629758548915677350851613, 1.00958751302124253563761879127, 1.77263646420677980862884008897, 3.51806011294757786392347036976, 3.93040239074190205832488305015, 4.62776441898860035357898879527, 5.87813113095379794617133851337, 6.87792052793118570192405516541, 7.42562264000474507715618978506, 8.528273703360519890398857964619, 9.401227086439273409389217932574

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