Properties

Label 2-1014-39.5-c1-0-7
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: $\chi_{1014} (239, \cdot )$
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

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

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