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Properties

Label 2-403-403.25-c1-0-30
Degree $2$
Conductor $403$
Sign $-0.932 + 0.360i$
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  − 0.921i·2-s + (−0.264 − 0.458i)3-s + 1.15·4-s + (−2.89 − 1.67i)5-s + (−0.422 + 0.243i)6-s + (−0.336 + 0.194i)7-s − 2.90i·8-s + (1.35 − 2.35i)9-s + (−1.53 + 2.66i)10-s + (1.48 + 0.858i)11-s + (−0.304 − 0.527i)12-s + (−2.58 + 2.51i)13-s + (0.178 + 0.309i)14-s + 1.76i·15-s − 0.372·16-s + (−3.39 − 5.88i)17-s + ⋯
L(s)  = 1  − 0.651i·2-s + (−0.152 − 0.264i)3-s + 0.575·4-s + (−1.29 − 0.747i)5-s + (−0.172 + 0.0995i)6-s + (−0.127 + 0.0734i)7-s − 1.02i·8-s + (0.453 − 0.785i)9-s + (−0.486 + 0.843i)10-s + (0.448 + 0.258i)11-s + (−0.0879 − 0.152i)12-s + (−0.717 + 0.696i)13-s + (0.0478 + 0.0828i)14-s + 0.456i·15-s − 0.0931·16-s + (−0.823 − 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.932 + 0.360i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185980 - 0.996962i\)
\(L(\frac12)\) \(\approx\) \(0.185980 - 0.996962i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.58 - 2.51i)T \)
31 \( 1 + (-1.53 + 5.35i)T \)
good2 \( 1 + 0.921iT - 2T^{2} \)
3 \( 1 + (0.264 + 0.458i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.89 + 1.67i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.336 - 0.194i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.48 - 0.858i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.39 + 5.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.04 - 1.75i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.13T + 23T^{2} \)
29 \( 1 - 6.43T + 29T^{2} \)
37 \( 1 + (-8.23 + 4.75i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.23 + 1.29i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0134 + 0.0233i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.52iT - 47T^{2} \)
53 \( 1 + (-5.34 + 9.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.19 + 1.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 9.90T + 61T^{2} \)
67 \( 1 + (1.42 + 0.824i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.4 - 7.75i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.522 - 0.301i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.05 + 1.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.4 - 8.33i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 - 1.89iT - 97T^{2} \)
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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

−11.36149243650128081021332275222, −9.907124235079856420122941563083, −9.258767912980898860835933540656, −7.995206555983151326964186034121, −7.08643577143468866489708700461, −6.35938885471469411703345663037, −4.51439178646184881034819064510, −3.88850313663144914465227915540, −2.31543917302922507031812641762, −0.65422167008930650568934824176, 2.39204585982989103879772296722, 3.78154848064537761573805957145, 4.85206885143091527829782566051, 6.29175201194168183290371171871, 6.93660766335402116336408459650, 7.952807743293704930969194433765, 8.382499752053363177031120577993, 10.26140595656567860556338977265, 10.69284267166906362228752423147, 11.55808161126700427915820485226

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