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Properties

Label 2-1850-5.4-c1-0-14
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 − 3.13i·3-s − 4-s + 3.13·6-s + 4.13i·7-s − i·8-s − 6.81·9-s − 3.68·11-s + 3.13i·12-s + 0.132i·13-s − 4.13·14-s + 16-s − 5.13i·17-s − 6.81i·18-s − 0.451·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.80i·3-s − 0.5·4-s + 1.27·6-s + 1.56i·7-s − 0.353i·8-s − 2.27·9-s − 1.10·11-s + 0.904i·12-s + 0.0367i·13-s − 1.10·14-s + 0.250·16-s − 1.24i·17-s − 1.60i·18-s − 0.103·19-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

\(L(1)\) \(\approx\) \(1.251827397\)
\(L(\frac12)\) \(\approx\) \(1.251827397\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.13iT - 3T^{2} \)
7 \( 1 - 4.13iT - 7T^{2} \)
11 \( 1 + 3.68T + 11T^{2} \)
13 \( 1 - 0.132iT - 13T^{2} \)
17 \( 1 + 5.13iT - 17T^{2} \)
19 \( 1 + 0.451T + 19T^{2} \)
23 \( 1 - 4.26iT - 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 - 5.58T + 31T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 5.07iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 + 4.39T + 61T^{2} \)
67 \( 1 + 0.228iT - 67T^{2} \)
71 \( 1 - 9.49T + 71T^{2} \)
73 \( 1 + 3.54iT - 73T^{2} \)
79 \( 1 - 0.903T + 79T^{2} \)
83 \( 1 - 13.9iT - 83T^{2} \)
89 \( 1 + 0.777T + 89T^{2} \)
97 \( 1 - 0.638iT - 97T^{2} \)
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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

−8.942340354704766337302877499000, −8.206787955992140033867067387006, −7.77656225226822835897681648141, −6.93376849671936932540703894497, −6.19691369194176909315497872852, −5.57960240456524738146679263740, −4.86113533550049109457613634772, −2.80021561467519197821358914369, −2.48979122939384225606567555788, −0.956512961306164535443886706869, 0.59134992460358639107036606780, 2.53446196530576023638673568231, 3.46672394732730356669730035858, 4.25024716455997960448699638693, 4.64601734439768776949004138676, 5.61397773674433921427064951460, 6.72204126564194028131659183909, 8.117863292978676444321367184458, 8.425626968499509009520572375573, 9.593049371676728454911868785571

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