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Properties

Label 2-1014-39.8-c1-0-49
Degree $2$
Conductor $1014$
Sign $-0.767 - 0.641i$
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 + (0.352 − 1.69i)3-s − 1.00i·4-s + (−0.499 + 0.499i)5-s + (−0.949 − 1.44i)6-s + (−1.39 + 1.39i)7-s + (−0.707 − 0.707i)8-s + (−2.75 − 1.19i)9-s + 0.705i·10-s + (−3.39 − 3.39i)11-s + (−1.69 − 0.352i)12-s + 1.97i·14-s + (0.670 + 1.02i)15-s − 1.00·16-s − 4.38·17-s + (−2.79 + 1.09i)18-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.203 − 0.979i)3-s − 0.500i·4-s + (−0.223 + 0.223i)5-s + (−0.387 − 0.591i)6-s + (−0.528 + 0.528i)7-s + (−0.250 − 0.250i)8-s + (−0.916 − 0.398i)9-s + 0.223i·10-s + (−1.02 − 1.02i)11-s + (−0.489 − 0.101i)12-s + 0.528i·14-s + (0.173 + 0.263i)15-s − 0.250·16-s − 1.06·17-s + (−0.657 + 0.259i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.767 - 0.641i$
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: \((2,\ 1014,\ (\ :1/2),\ -0.767 - 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7287589730\)
\(L(\frac12)\) \(\approx\) \(0.7287589730\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.352 + 1.69i)T \)
13 \( 1 \)
good5 \( 1 + (0.499 - 0.499i)T - 5iT^{2} \)
7 \( 1 + (1.39 - 1.39i)T - 7iT^{2} \)
11 \( 1 + (3.39 + 3.39i)T + 11iT^{2} \)
17 \( 1 + 4.38T + 17T^{2} \)
19 \( 1 + (-1.70 - 1.70i)T + 19iT^{2} \)
23 \( 1 - 0.998T + 23T^{2} \)
29 \( 1 + 0.998iT - 29T^{2} \)
31 \( 1 + (6.50 + 6.50i)T + 31iT^{2} \)
37 \( 1 + (4.10 - 4.10i)T - 37iT^{2} \)
41 \( 1 + (-5.24 + 5.24i)T - 41iT^{2} \)
43 \( 1 + 8.88iT - 43T^{2} \)
47 \( 1 + (0.352 + 0.352i)T + 47iT^{2} \)
53 \( 1 + 14.2iT - 53T^{2} \)
59 \( 1 + (-0.998 - 0.998i)T + 59iT^{2} \)
61 \( 1 + 9.59T + 61T^{2} \)
67 \( 1 + (-5.79 - 5.79i)T + 67iT^{2} \)
71 \( 1 + (7.13 - 7.13i)T - 71iT^{2} \)
73 \( 1 + (-1.70 + 1.70i)T - 73iT^{2} \)
79 \( 1 + 0.207T + 79T^{2} \)
83 \( 1 + (-9.17 + 9.17i)T - 83iT^{2} \)
89 \( 1 + (-2.54 - 2.54i)T + 89iT^{2} \)
97 \( 1 + (-3.58 - 3.58i)T + 97iT^{2} \)
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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

−9.334899314662406697538059460328, −8.667027853537051549920799499623, −7.71289087910827450650475225890, −6.86281197377835629861032740916, −5.89426874461424354677583052407, −5.33720956244941599177400810594, −3.71530577645530991042900417051, −2.91236993699083040535085492736, −2.00255853036477770416750878133, −0.24982158451112012885221036390, 2.47734595394730749128640337337, 3.49223497135434474538196890861, 4.54952633254237913824971892424, 4.94331727839183519207521483502, 6.12376421997132353936260744753, 7.15256464021678503690968195060, 7.86654227937522009822903609523, 8.890713295445387152864342933976, 9.537190629097432489493449257147, 10.51837542590854547216902920792

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