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Properties

Label 2-403-13.10-c1-0-0
Degree $2$
Conductor $403$
Sign $-0.555 + 0.831i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
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  + (−2.18 + 1.25i)2-s + (−0.435 − 0.754i)3-s + (2.17 − 3.76i)4-s + 3.48i·5-s + (1.90 + 1.09i)6-s + (−2.02 − 1.17i)7-s + 5.90i·8-s + (1.12 − 1.93i)9-s + (−4.39 − 7.60i)10-s + (−2.90 + 1.67i)11-s − 3.78·12-s + (3.60 + 0.190i)13-s + 5.89·14-s + (2.63 − 1.51i)15-s + (−3.09 − 5.36i)16-s + (−3.20 + 5.54i)17-s + ⋯
L(s)  = 1  + (−1.54 + 0.890i)2-s + (−0.251 − 0.435i)3-s + (1.08 − 1.88i)4-s + 1.55i·5-s + (0.776 + 0.448i)6-s + (−0.766 − 0.442i)7-s + 2.08i·8-s + (0.373 − 0.646i)9-s + (−1.38 − 2.40i)10-s + (−0.876 + 0.506i)11-s − 1.09·12-s + (0.998 + 0.0527i)13-s + 1.57·14-s + (0.679 − 0.392i)15-s + (−0.773 − 1.34i)16-s + (−0.776 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.555 + 0.831i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.555 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0104224 - 0.0195033i\)
\(L(\frac12)\) \(\approx\) \(0.0104224 - 0.0195033i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.60 - 0.190i)T \)
31 \( 1 + iT \)
good2 \( 1 + (2.18 - 1.25i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.435 + 0.754i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.48iT - 5T^{2} \)
7 \( 1 + (2.02 + 1.17i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.90 - 1.67i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.20 - 5.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.21 + 1.27i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.899 + 1.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.28 + 7.42i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (7.27 - 4.20i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.15 + 2.97i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.63 + 2.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.36iT - 47T^{2} \)
53 \( 1 + 9.35T + 53T^{2} \)
59 \( 1 + (11.1 + 6.45i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.79 - 3.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.992 - 0.573i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.59 + 4.96i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.97iT - 73T^{2} \)
79 \( 1 - 1.26T + 79T^{2} \)
83 \( 1 - 7.88iT - 83T^{2} \)
89 \( 1 + (-8.94 + 5.16i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0440 + 0.0254i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.29367263941443384720640216895, −10.52676801772283523815186498517, −10.14667861474497858437862971739, −9.096583611845706503074385749089, −7.952675499581440535164813140431, −7.18763903646157839146370994756, −6.38422358150454856997455398541, −6.17181201290301444175220540021, −3.73917956879822382796822928017, −2.02455591561676987837419208952, 0.02379923235721282620129966117, 1.62309420283561579535090689348, 3.11771011086260893002248294020, 4.63443401783791468111075603353, 5.77180336012776970332788219419, 7.42896398770013392857580519272, 8.350812842252268378974512391414, 9.101549605861574976558216739136, 9.519323013197943005488658836639, 10.70576872560412147433415727231

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