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Properties

Label 2-1014-39.8-c1-0-34
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  + (0.707 − 0.707i)2-s + (−1.13 + 1.30i)3-s − 1.00i·4-s + (0.0594 − 0.0594i)5-s + (0.124 + 1.72i)6-s + (3.26 − 3.26i)7-s + (−0.707 − 0.707i)8-s + (−0.429 − 2.96i)9-s − 0.0841i·10-s + (−1.35 − 1.35i)11-s + (1.30 + 1.13i)12-s − 4.61i·14-s + (0.0104 + 0.145i)15-s − 1.00·16-s − 6.93·17-s + (−2.40 − 1.79i)18-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.654 + 0.756i)3-s − 0.500i·4-s + (0.0266 − 0.0266i)5-s + (0.0507 + 0.705i)6-s + (1.23 − 1.23i)7-s + (−0.250 − 0.250i)8-s + (−0.143 − 0.989i)9-s − 0.0266i·10-s + (−0.408 − 0.408i)11-s + (0.378 + 0.327i)12-s − 1.23i·14-s + (0.00270 + 0.0375i)15-s − 0.250·16-s − 1.68·17-s + (−0.566 − 0.423i)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: $\chi_{1014} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ -0.397 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.420411955\)
\(L(\frac12)\) \(\approx\) \(1.420411955\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (1.13 - 1.30i)T \)
13 \( 1 \)
good5 \( 1 + (-0.0594 + 0.0594i)T - 5iT^{2} \)
7 \( 1 + (-3.26 + 3.26i)T - 7iT^{2} \)
11 \( 1 + (1.35 + 1.35i)T + 11iT^{2} \)
17 \( 1 + 6.93T + 17T^{2} \)
19 \( 1 + (-0.229 - 0.229i)T + 19iT^{2} \)
23 \( 1 + 0.747T + 23T^{2} \)
29 \( 1 + 9.96iT - 29T^{2} \)
31 \( 1 + (3.78 + 3.78i)T + 31iT^{2} \)
37 \( 1 + (-3.42 + 3.42i)T - 37iT^{2} \)
41 \( 1 + (1.08 - 1.08i)T - 41iT^{2} \)
43 \( 1 + 7.63iT - 43T^{2} \)
47 \( 1 + (-5.42 - 5.42i)T + 47iT^{2} \)
53 \( 1 + 5.64iT - 53T^{2} \)
59 \( 1 + (5.89 + 5.89i)T + 59iT^{2} \)
61 \( 1 - 5.98T + 61T^{2} \)
67 \( 1 + (-2.57 - 2.57i)T + 67iT^{2} \)
71 \( 1 + (4.36 - 4.36i)T - 71iT^{2} \)
73 \( 1 + (-6.67 + 6.67i)T - 73iT^{2} \)
79 \( 1 - 6.25T + 79T^{2} \)
83 \( 1 + (8.88 - 8.88i)T - 83iT^{2} \)
89 \( 1 + (-8.61 - 8.61i)T + 89iT^{2} \)
97 \( 1 + (5.34 + 5.34i)T + 97iT^{2} \)
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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

−9.969462358411382828136401173032, −9.111806364203873546444548579887, −8.038088582516848199524381574511, −7.05998450047624987283073903870, −6.00246488572104158508728527282, −5.10116519589489857624037318739, −4.32670044085270308945341191998, −3.76963170582682674369931418856, −2.13526773081308463471987310419, −0.58277946536536250578283592698, 1.79124504423393494730299712103, 2.64946861702975902118400445911, 4.56941279712046383152621712710, 5.06005566484234269659517253788, 5.92107886988497210094175560728, 6.75130092869853431462274021376, 7.57670273504026407323387019618, 8.450286996659592033520453579056, 8.970220960232010414683735796113, 10.55889167614356415248459286133

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