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Properties

Label 2-507-13.10-c1-0-19
Degree $2$
Conductor $507$
Sign $-0.565 + 0.824i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (−0.358 + 0.207i)2-s + (−0.5 − 0.866i)3-s + (−0.914 + 1.58i)4-s + 2.82i·5-s + (0.358 + 0.207i)6-s + (−2.44 − 1.41i)7-s − 1.58i·8-s + (−0.499 + 0.866i)9-s + (−0.585 − 1.01i)10-s + (−1.73 + i)11-s + 1.82·12-s + 1.17·14-s + (2.44 − 1.41i)15-s + (−1.49 − 2.59i)16-s + (3.82 − 6.63i)17-s − 0.414i·18-s + ⋯
L(s)  = 1  + (−0.253 + 0.146i)2-s + (−0.288 − 0.499i)3-s + (−0.457 + 0.791i)4-s + 1.26i·5-s + (0.146 + 0.0845i)6-s + (−0.925 − 0.534i)7-s − 0.560i·8-s + (−0.166 + 0.288i)9-s + (−0.185 − 0.320i)10-s + (−0.522 + 0.301i)11-s + 0.527·12-s + 0.313·14-s + (0.632 − 0.365i)15-s + (−0.374 − 0.649i)16-s + (0.928 − 1.60i)17-s − 0.0976i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.565 + 0.824i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.565 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0696894 - 0.132247i\)
\(L(\frac12)\) \(\approx\) \(0.0696894 - 0.132247i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (0.358 - 0.207i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 2.82iT - 5T^{2} \)
7 \( 1 + (2.44 + 1.41i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.73 - i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.82 + 6.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.44 + 1.41i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.17iT - 31T^{2} \)
37 \( 1 + (6.63 - 3.82i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.47 - 2.58i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.828 - 1.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (6.63 + 3.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.65 - 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.91 - 3.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.73 - i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.343iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 3.65iT - 83T^{2} \)
89 \( 1 + (-12.8 + 7.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.16 - 1.82i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.36027591539744734199767289949, −9.947252379662204761790576657244, −8.734467763028337462379704637093, −7.56473105674897897438632480416, −7.09397359001931096436689265493, −6.37754927249453332463026334684, −4.88044038463577459346636162154, −3.45799338825099757015389538539, −2.70594363963673850540200614835, −0.10042942248156386499196955596, 1.58678236435732339620495935140, 3.54691672600107907307569238146, 4.72179981384031546063990436712, 5.66846121594211652997725172933, 6.10705894757318044911800926544, 8.010994190356005153901181346513, 8.813076974873225259396555933363, 9.439879408669720101520533745005, 10.23029744385223792163173299141, 10.91532606475456443670479207529

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