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Properties

Label 2-507-39.5-c1-0-33
Degree $2$
Conductor $507$
Sign $-0.172 + 0.985i$
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  + (0.540 + 0.540i)2-s + (0.0858 − 1.72i)3-s − 1.41i·4-s + (−0.996 − 0.996i)5-s + (0.981 − 0.888i)6-s + (1.80 + 1.80i)7-s + (1.84 − 1.84i)8-s + (−2.98 − 0.296i)9-s − 1.07i·10-s + (3.35 − 3.35i)11-s + (−2.44 − 0.121i)12-s + 1.94i·14-s + (−1.80 + 1.63i)15-s − 0.837·16-s − 5.80·17-s + (−1.45 − 1.77i)18-s + ⋯
L(s)  = 1  + (0.382 + 0.382i)2-s + (0.0495 − 0.998i)3-s − 0.708i·4-s + (−0.445 − 0.445i)5-s + (0.400 − 0.362i)6-s + (0.681 + 0.681i)7-s + (0.652 − 0.652i)8-s + (−0.995 − 0.0989i)9-s − 0.340i·10-s + (1.01 − 1.01i)11-s + (−0.707 − 0.0350i)12-s + 0.520i·14-s + (−0.467 + 0.422i)15-s − 0.209·16-s − 1.40·17-s + (−0.342 − 0.417i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.172 + 0.985i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.172 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07019 - 1.27330i\)
\(L(\frac12)\) \(\approx\) \(1.07019 - 1.27330i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0858 + 1.72i)T \)
13 \( 1 \)
good2 \( 1 + (-0.540 - 0.540i)T + 2iT^{2} \)
5 \( 1 + (0.996 + 0.996i)T + 5iT^{2} \)
7 \( 1 + (-1.80 - 1.80i)T + 7iT^{2} \)
11 \( 1 + (-3.35 + 3.35i)T - 11iT^{2} \)
17 \( 1 + 5.80T + 17T^{2} \)
19 \( 1 + (2.39 - 2.39i)T - 19iT^{2} \)
23 \( 1 - 3.39T + 23T^{2} \)
29 \( 1 + 6.57iT - 29T^{2} \)
31 \( 1 + (0.386 - 0.386i)T - 31iT^{2} \)
37 \( 1 + (-5.93 - 5.93i)T + 37iT^{2} \)
41 \( 1 + (-0.734 - 0.734i)T + 41iT^{2} \)
43 \( 1 - 7.56iT - 43T^{2} \)
47 \( 1 + (0.243 - 0.243i)T - 47iT^{2} \)
53 \( 1 + 2.07iT - 53T^{2} \)
59 \( 1 + (-3.56 + 3.56i)T - 59iT^{2} \)
61 \( 1 - 7.04T + 61T^{2} \)
67 \( 1 + (-4.54 + 4.54i)T - 67iT^{2} \)
71 \( 1 + (-6.79 - 6.79i)T + 71iT^{2} \)
73 \( 1 + (-6.04 - 6.04i)T + 73iT^{2} \)
79 \( 1 - 8.77T + 79T^{2} \)
83 \( 1 + (-8.31 - 8.31i)T + 83iT^{2} \)
89 \( 1 + (9.62 - 9.62i)T - 89iT^{2} \)
97 \( 1 + (-1.34 + 1.34i)T - 97iT^{2} \)
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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

−11.20918490677363770840074756354, −9.563672071486038214112358352584, −8.573497800825318320949905679213, −8.091328351051846168509684858951, −6.64324421108824689061794747695, −6.22523475683991554061285506349, −5.14724066052989773099250351900, −4.08573909154597680122091932197, −2.24860497541870972387626975630, −0.924827174170892772134276204563, 2.25139628679092105703615216932, 3.61879626395165051695893851858, 4.28027224039117239279979872987, 4.95731488633839409988485623828, 6.79084536823466859276440563068, 7.47851727349917443070084542353, 8.671229600494020044419187541827, 9.305809191818198624307234850231, 10.75224723117033862847861865822, 11.01863259869502721365809753813

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