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Properties

Label 2-1014-13.10-c1-0-7
Degree $2$
Conductor $1014$
Sign $0.397 - 0.917i$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
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  + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + 4.04i·5-s + (0.866 + 0.499i)6-s + (0.599 + 0.346i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−2.02 − 3.50i)10-s + (4.20 − 2.42i)11-s − 0.999·12-s − 0.692·14-s + (3.50 − 2.02i)15-s + (−0.5 − 0.866i)16-s + (3.69 − 6.39i)17-s − 0.999i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + 1.81i·5-s + (0.353 + 0.204i)6-s + (0.226 + 0.130i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.640 − 1.10i)10-s + (1.26 − 0.731i)11-s − 0.288·12-s − 0.184·14-s + (0.905 − 0.522i)15-s + (−0.125 − 0.216i)16-s + (0.895 − 1.55i)17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.397 - 0.917i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.397 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.183188265\)
\(L(\frac12)\) \(\approx\) \(1.183188265\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good5 \( 1 - 4.04iT - 5T^{2} \)
7 \( 1 + (-0.599 - 0.346i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.20 + 2.42i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.69 + 6.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.54 - 0.890i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.55 - 4.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.67 - 2.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.972iT - 31T^{2} \)
37 \( 1 + (1.11 - 0.643i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.30 + 0.753i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.15 - 7.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.20iT - 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + (-1.13 - 0.653i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.198 + 0.343i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.24 + 3.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.15 - 0.664i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.65iT - 73T^{2} \)
79 \( 1 + 8.33T + 79T^{2} \)
83 \( 1 - 15.3iT - 83T^{2} \)
89 \( 1 + (2.69 - 1.55i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.39 - 4.27i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.01777361670028664091890750003, −9.406527908203005277022684122945, −8.290889193122408985307612416953, −7.30960929541492096013785861148, −6.95162500956477805983086900518, −6.14592225692760588159727832846, −5.29904976037512535832150718356, −3.53590429812047900045684543870, −2.71050148869356301557517713413, −1.21241266981680682007122827477, 0.855631390080655018313249176957, 1.80665703023923259608772885903, 3.77052111751258141526194208071, 4.39792571530294924561230569565, 5.32539101705227932224959995377, 6.34531973520067041201051457854, 7.55129644660961981944032476356, 8.604299164203192275687038010829, 8.852888322860286036395229966157, 9.820047825993543621513853158576

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